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ISSN 1937 - 1055

VOLUME 4, 2019

INTERNATIONAL JOURNAL OF

MATHEMATICAL COMBINATORICS

EDITED BY

THE MADIS OF CHINESE ACADEMY OF SCIENCES AND

ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA

December, 2019

Vol.4, 2019 ISSN 1937-1055

International Journal of

Mathematical Combinatorics

(www.mathcombin.com)

Edited By

The Madis of Chinese Academy of Sciences and

Academy of Mathematical Combinatorics & Applications, USA

December, 2019

Aims and Scope: The mathematical combinatorics is a subject that applying combinatorial

notion to all mathematics and all sciences for understanding the reality of things in the universe,

motivated by CC Conjecture of Dr.Linfan MAO on mathematical sciences. The International

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Mathematical combinatorics;

Smarandache multi-spaces and Smarandache geometries with applications to other sciences;

Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and

map enumeration; Combinatorial designs; Combinatorial enumeration;

Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential

Topology; Topology of Manifolds;

Geometrical aspects of Mathematical Physics and Relations with Manifold Topology;

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Editor-in-Chief

Linfan MAO

Chinese Academy of Mathematics and System

Science, P.R.China

and

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Beijing University of Civil Engineering and

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Jadavpur University, India


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Hassan II University Mohammedia

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Junliang Cai

Beijing Normal University, P.R.China


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Beijing Jiaotong University, P.R.China


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Beijing University of Civil Engineering and

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Capital Normal University, P.R.China


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Hunan Normal University, P.R.China


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Indian Institute of Technology, India


Ion Patrascu

Fratii Buzesti National College

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Han Ren

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Ovidiu-Ilie Sandru

Politechnica University of Bucharest

Romania

ii International Journal of Mathematical Combinatorics

Mingyao Xu

Peking University, P.R.China


Guiying Yan


Science, P.R.China


Y. Zhang

Department of Computer Science

Georgia State University, Atlanta, USA

Famous Words:

If you want to know everything all at once, you will know nothing.

By Ivan Pavlov, a former Soviet physiologist, psychologist.

International J.Math. Combin. Vol.4(2019), 1-18

Graphs, Networks and Natural Reality

– from Intuitive Abstracting to Theory

Linfan MAO

1. Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China

2. Academy of Mathematical Combinatorics & Applications (AMCA), Colorado, USA

E-mail: [email protected]

Abstract: In the view of modern science, a matter is nothing else but a complex network−→G , i.e., the reality of matter is characterized by complex network. However, there are no

such a mathematical theory on complex network unless local and statistical results. Could we

establish such a mathematics on complex network? The answer is affirmative, i.e., mathemat-

ical combinatorics or mathematics over topological graphs. Then, what is a graph? How does

it appears in the universe? And what is its role for understanding of the reality of matters?

The main purpose of this paper is to survey the progressing process and explains the notion

from graphs to complex network and then, abstracts mathematical elements for understand-

ing reality of matters. For example, L.Euler’s solving on the problem of Konigsberg seven

bridges resulted in graph theory and embedding graphs in compact n-manifold, particularly,

compact 2-manifold or surface with combinatorial maps and then, complex networks with

reality of matters. We introduce 2 kinds of mathematical elements respectively on living

body or non-living body for self-adaptive systems in the universe, i.e., continuity flow and

harmonic flow−→GL which are essentially elements in Banach space over graphs with operator

actions on ends of edges in graph−→G . We explain how to establish mathematics on the 2

kinds of elements, i.e., vectors underling a combinatorial structure−→G by generalize a few

well-known theorems on Banach or Hilbert space and contribute mathematics on complex

networks. All of these imply that graphs expand the mathematical field, establish the foun-

dation on holding on the nature and networks are closer more to the real but without a

systematic theory. However, its generalization enables one to establish mathematics over

graphs, i.e., mathematical combinatorics on reality of matters in the universe.

Key Words: Graph, 2-cell embedding of graph, combinatorial map, complex network,

reality, mathematical element, Smarandache multispace, mathematical combinatorics.

AMS(2010): 00A69,05C21,05C25,05C30 05C82, 15A03,57M20

§1. Introduction

What is the role of mathematics to natural reality? Certainly, as the science of quantity,

mathematics is the main tool for humans understanding matters, both for the macro and the

micro in the universe. Generally, it builds a model and characterizes the behavior of a matter

1Received August 25, 2019, Accepted November 20, 2019.

2 Linfan MAO

for holding on reality and then, establishes a theory, such as those shown in Fig.1.�� - - -Observation& Experiment

Induction& Deduction

Hypothesis& Testing Theory

Fig.1

This scientific method on matters in the universe is completely reflected in the solving

process of L.Euler on the problem of Konigsberg seven bridges. Geographically, the city of

Konigsberg is located on both sides of Pregel River, including two large islands which were

connected to each other and the mainland by seven bridges, such as those shown in Fig.2.

The residents of Konigsberg usually wished to pass through each bridge once without repeat,

initialing at point of the mainland or islands.

Fig.2

However, no one traveled in such a way once. Then, a resident should how to travel for such a

walk? L.Euler solved this problem, and answered it had no solution in 1736. How did he do it?

Let A,B,C,D be the two sides and islands. Then, he abstracted this problem on (a) equivalent

to finding a traveling passing through each lines on (b) without repeating.

(a) (b)

Fig.3

Clearly, such a traveling must be with the same in and out times at each point A,B,C or D.

But, (b) is not fitted with such conditions. So, there are no such a traveling in the problem on

Graphs, Networks and Natural Reality – from Intuitive Abstracting to Theory 3

Konigsberg seven bridges.

Euler’s solving method on the problem of Konigsberg seven bridges finally resulted graph

theory into beings today. A graph G is an ordered 3-tuple (V,E; I), where V,E are finite sets,

V 6= ∅ and I : E → V × V . Call V the vertex set and E the edge set of G, denoted by V (G)

and E(G), respectively. For example, two graphs K(3, 3) and K5 are shown in Fig.4.r r rr r r r r rr rK(3, 3) K5

Fig.4

Usually, if (u, v) = (v, u) for ∀u, v ∈ V (G), then G is called a graph. Otherwise, it is called

a directed graph with an orientation u→ v on each edge (u, v), denoted by−→G .

Let G1 = (V1, E1, I1), G2 = (V2, E2, I2) be 2 graphs. If there exists a 1 − 1 mapping

φ : V1 → V2 and φ : E1 → E2 such that φI1(e) = I2φ(e) for ∀e ∈ E1 with the convention that

φ(u, v) = (φ(u), φ(v)), then we say that G1 is isomorphic to G2, denoted by G1∼= G2 and φ

an isomorphism between G1 and G2. Clearly, all automorphisms φ : V (G) → V (G) of graph G

form a group under the composition operation, and denoted by AutG the automorphism group

of graph G. A few automorphism groups of well-known graphs are listed in Table 1.

G AutG order

Pn Z2 2

Cn Dn 2n

Kn Sn n!

Km,n(m 6= n) Sm × Sn m!n!

Kn,n S2[Sn] 2n!2

Table 1

Certainly, an edge e = uv ∈ E(G) can be divided into two semi-arcs eu, ev such as those

shown in Fig.5.u uu v

- �u uu v

- eu eve Divided into

Fig.5

Similarly, two semi-arcs eu, fv are called v-incident or e-incident if u = v or e = f . Denote

all semi-arcs of a graph G by X 12(G). A 1−1 mapping ξ on X 1

2(G) such that ∀eu, fv ∈ X 1

2(G),

ξ(eu) and ξ(fv) are v−incident or e−incident if eu and fv are v−incident or e−incident, is

4 Linfan MAO

called a semi-arc automorphism of the graph G. Clearly, all semi-arc automorphisms of a graph

also form a group, denoted by Aut 12G.

Certainly, graph theory studies properties of graphs. A property is nothing else but a

family of graph, i.e., P = {G1, G2, · · · , Gn, · · · } but closed under isomorphisms φ of graphs,

i.e., Gφ ∈ P if G ∈ P. For example, hamiltonian graphs, Euler graphs and also interesting

parameters, such as those of connectivity, independent number, covering number, girth, level

number, · · · of a graph.

The main purpose of this paper is to survey the progressing process and explains the notion

from graphs to complex network and then, abstracts mathematical elements for understanding

reality of matters. For example, L.Euler’s solving on the problem of Konigsberg seven bridges

resulted in graph theory and embedding graphs in compact n-manifold, particularly, compact

2-manifold or surface with combinatorial maps and then, complex networks with reality of

matters. We introduce 2 kinds of mathematical elements respectively on living or non-living

body in the universe, i.e., continuity and harmonic flows−→GL which are essentially elements

in Banach space over graphs with operator actions on ends of edges in graph−→G . We explain

how to establish mathematics on the 2 kinds of elements, i.e., vectors underling a combinatorial

structure−→G by generalize a few well-known theorems on Banach or Hilbert space and contribute

a mathematics on complex networks.

For terminologies and notations not mentioned here, we follow references [1],[2] and [4] for

graphs, [3] for complex network, [6] for automorphisms of graph, [24] for algebraic topology,

[25] for elementary particles and [6],[26] for Smarandache systems and multispaces.

§2. Embedding Graphs on Surfaces

2.1 Surface

A surface is a 2-dimensional compact manifold without boundary. For example, a few surfaces

are shown in Fig.6.

Sphere Torus Klein bottle

Fig.6

Clearly, the intuition imagination is difficult for determining surface of higher genus. How-

ever, T.Rado showed the following result, which is the fundamental of combinatorial topology,


ro topological graphs on surfaces.

Theorem 2.1(T.Rado 1925,[24]) For any compact surface S, there exist a triangulation {Ti, i ≥1} on S.

T.Rado’s result on triangulation of surface enables one to present a surface by listing every

triangle with each side a label and a direction, i.e., the polygon representation. Then, the surface

is assembled by identifying the two sides with the same label and direction. This way results

in a polygon representation on a surface finally. For examples, the polygon representations on

surfaces in Fig.6 are shown in Fig.7.

��- 6 6 6-

---6 ?a a

B

A

a a

b

b

b

b

a a

Sphere Torus Klein bottle

Fig.7

We know the classification theorem of surfaces following.

Theorem 2.2([24]) Any connected compact surface S is either homeomorphic to a sphere, or to

a connected sum of tori, or to a connected sum of projective planes, i.e., its surface presentation

S is elementary equivalent to one of the standard surface presentations following:

(1) The sphere S2 =⟨a|aa−1

⟩;

(2) The connected sum of p tori

T 2#T 2# · · ·#T 2

︸︷︷︸p

=

⟨ai, bi, 1 ≤ i ≤ p |

p∏

i=1

aibia−1i b−1

i

⟩;

(3) The connected sum of q projective planes

P 2#P 2 · · ·#P 2

︸︷︷︸q

=

⟨ai, 1 ≤ i ≤ q |

q∏

i=1

ai

⟩.

A combinatorial proof on Theorem 2.2 can be found in [6]. By definition, the Euler

characteristic of S is

χ(S) = |V (S)| − |E(S)| + |F (S)|,

where V (S), E(S) and F (S) are respective the set of vertex set, edge set and face set of the

polygon representation of surface S. Then, we know the next result.

6 Linfan MAO

Theorem 2.3([24]) Let S be a connected compact surface with a presentation S. Then

χ(S) =

2, if S ∼El S2,

2 − 2p, if S ∼El T2#T 2# · · ·#T 2

︸︷︷︸p

,

2 − q, if S ∼El P2#P 2# · · ·#P 2

︸︷︷︸q

.

Theorem 2.3 enables one to define the genus of orientable or non-orientable surface S by

numbers p and q, respectively, and the genus of sphere is defined to be 0.

2.2 Embedding Graph

A graph G is said to be embeddable into a topological space T if there is a 1 − 1 continuous

mapping φ : G → T with φ(p) 6= φ(q) if p, q are different points on graph G. Particularly, if

T = R2 is a Euclidean plane, we say that G is a planar graph.

A most interesting case on the embedding problem of graphs is the case of surface, which is

essentially to search the polyhedral structures on surfaces. Clearly, many results on embedding

graphs is on these surfaces with small genus. For example, embedding results on p = 0 the

sphere, p = 1 the torus, · · · of orientable surfaces, or on q = 1 the projective plane, q = 2

the Kein bottle, · · · of non-orientable surfaces. The most simple case is embedding graphs on

sphere which is equivalent to a planar graph, such as the dodecahedron shown in Fig.8.

Dodecahedron Planar graph

Fig.8

We have known a few criterions on planar graphs following.

Theorem 2.4(Euler,1758, [2]) Let G be a planar graph with p vertices, q edges and r faces.

Then, p− q + r = 2.

Theorem 2.5(Kuratowski,1930, [1]) A graph is planar if and only if it contains no subgraphs

homeomorphic with K5 or K(3, 3).

A 2-cell embedding of G on surface S is defined to be a continuous 1-1 mapping τ : G→ S

such that each component in S \ τ(G) homeomorphic to an open 2-disk { (x, y) | x2 + y2 < 1}.Certainly, the image τ(G) is contained in the 1-skeleton of a triangulation on the surface S.


For example, the embedding of K4 on the sphere and Klein bottle are shown in Fig.9 (a) and

(b), respectively.

bbbrrr r r r rr1 1

2

2

u1

u2

u3 u4

u1

u2u3

u46 6-�

Fig.9

There is an algebraic representation for characterizing the 2-cell embedding of graphs. For

v ∈ V (G), denote by NeG(v) = {e1, e2, · · · , eρ(v)} all the edges incident with the vertex v. A

permutation on e1, e2, · · · , eρ(v) is said to be a pure rotation and all pure rotations incident

with v is denoted by (v). Generally, a pure rotation system of the graph G is defined to be

ρ(G) = {(v)|v ∈ V (G)} which was observed and used by Dyck in 1888, Heffter in 1891 and

then formalized by Edmonds in 1960. For example,

ρ(K4) = {(u1u4, u1u3, u1u2), (u2u1, u2u3, u2u4), (u3u1, u3u4, u3u2), (u4u1, u4u2, u4u3)},ρ(K4) = {(u1u2, u1u3, u1u4), (u2u1, u2u3, u2u4), (u3u2, u3u4, u3u1), (u4u1, u4u2, u4u3)}

are respectively the pure rotation systems for embeddings of K4 on the sphere and Klein bottle

shown in Fig.9.

Theorem 2.6(Heffter 1891, Edmonds 1960, [4]) Every embedding of a graph G on an orientable

surface S induces a unique pure rotation system ρ(G). Conversely, Every pure rotation system

ρ(G) of a graph G induces a unique embedding of G on an orientable surface S.

Clearly, an embedding of graph G can be associated 0, 1 or 2-band respectively with

vertices, edges and face on its surface. A band decomposition is called locally orientable if each

0-band is assigned an orientation, and a 1-band is called orientation-preserving if the direction

induced on its ends by adjoining 0-bands are the same as those induced by one of the two

possible orientations of the 1-band. Otherwise, orientation-reversing. An edge e in a graph G

embedded on a surface S associated with a locally orientable band decomposition is said to be

type 0 if its corresponding 1-band is orientation-preserving and otherwise, type 1. A rotation

system ρL(v) of v ∈ V (G) to be a pair (J (v), λ), where J (v) is a pure rotation system and

λ : E(G) → Z2 is determined by λ(e) = 0 or λ(e) = 1 if e is type 0 or type 1 edge, respectively.

Theorem 2.7(Ringel 1950s, Stahl 1978,[4]) Every rotation system on a graph G defines a

unique locally orientable 2-cell embedding of G → S. Conversely, every 2-cell embedding of a

graph G→ S defines a rotation system for G.

8 Linfan MAO

A 2-cell embedding of a connected graph on surface is called map by W.T.Tutte. He

characterized embeddings by purely algebra in 1973 ([5], [26]). By his definition, an embedding

M = (Xα,β ,P) is defined to be a basic permutation P , i.e, for any x ∈ Xα,β, no integer k exists

such that Pkx = αx, acting on Xα,β , the disjoint union of quadricells Kx of x ∈ X (the base

set), where K = {1, α, β, αβ} is the Klein group, satisfying the following two conditions:

(1) αP = P−1α;

(2) the group ΨJ = 〈α, β,P〉 is transitive on Xα,β .

Furthermore, if the group ΨI = 〈αβ,P〉 is transitive on Xα,β , then M is non-orientable.

Otherwise, orientable.

For example, the embedding (Xα,β ,P) of graph K4 on torus shown in Fig.10.

--

6 6�� ZZZ rr rr?�}+�1

2 2

1

x

y

z

u

v

wsFig.10

can be algebraic represented by

Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy,βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw},

P = (x, y, z)(αβx, u, w)(αβz, αβu, v)(αβy, αβv, αβw)

×(αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv),

with vertices

v1 = {(x, y, z), (αx, αz, αy)}, v2 = {(αβx, u, w), (βx, αw, αu)},v3 = {(αβz, αβu, v), (βz, αv, βu)}, v4 = {(αβy, αβv, αβw), (βy, βw, βv)},

edges

{e, αe, βe, αβe}, e ∈ {x, y, z, u, v, w}

and faces

f1 = {(x, u, v, αβw, αβx, y, αβv, αβz), (βx, αz, αv, βy, αx, αw, βv, βu)},f2 = {(z, αβu,w, αβy), (βz, αy, βw, αu)}.

Two embeddings M1 = (X 1α,β ,P1) and M2 = (X 2

α,β ,P2) are said to be isomorphic if there


exists a bijection ξ

ξ : X 1α,β −→ X 2

α,β

such that for ∀x ∈ X 1α,β , ξα(x) = αξ(x), ξβ(x) = βξ(x), ξP1(x) = P2ξ(x). Particularly, if M1 =

M2 = M , an isomorphism between M1 and M2 is then called an automorphism of embedding

M . Clearly, all automorphisms of a embedding M form a group, called the automorphism group

of M , denoted by AutM .

There are two main problems on embedding of graphs on surfaces following.

Problem 2.1 Let G be a graph and S a surface. Whether or not G can be embedded on S?

This problem had been solved by Duke on orientable case in 1966, and Stahl on non-

orientable case in 1978. They obtained the result following.

Theorem 2.8(Duke 1966, Stahl 1978,[4]) Let G be a connected graph and let GR(G), CR(G)

be the respective genus range of G on orientable or non-orientable surfaces. Then, GR(G) and

CR(G) both are unbroken interval of integers.

Theorem 2.8 bring about to determine the minimum and maximum genus γ(G), γM (G) of

graph G on surfaces. Among them, the most simple case is to determine the maximum genus

γM (G) on non-orientable case, which was obtained by Edmonds in 1965. It is the Betti number

β(G) = |E(G)|− |V (G)|+1. The maximum genus γM (G) of G on orientable case is determined

by Xuong with the deficiency ξ(G), i.e., the minimum number of components in G \ T for all

spanning trees T in G in 1979. However, it is difficult for the minimum genus γ(G), only a few

results on typical graphs. For example, the genus of Kn and Km,n are listed following.

Theorem 2.9(Ringel and Youngs 1968, [4]) The minimum genus of a complete graph is given

by

γ(Kn) =

⌈(n− 3)(n− 4)

12

⌉, n ≥ 3.

Theorem 2.10(Ringel 1965, [4]) The minimum genus of a complete bipartite graph is given by

γ(K(m,n)) =

⌈(m− 2)(n− 2)

4

⌉, m, n ≥ 2.

Problem 2.2 Let G be a graph and S a surface. How many non-isomorphic embeddings of G

on S?

This problem is difficult, only be partially solved until today. However, the following simple

result enables one to enumerate rooted embeddings, where an embedding (Xα,β ,P) is rooted

on an element r ∈ Xα,β if r is marked beforehand.

Theorem 2.11([5],[6]) The auotmorphism group of a rooted embedding M , i.e., AutM r is

trivial.

Theorem 2.12([5],[6]) |AutM | | |Xα,β | = 4ε(M).

10 Linfan MAO

A root r in an embedding M is called an i-root if it is incident to a vertex of valency i. Two

i-roots r1, r2 are transitive if there exists an automorphism τ ∈ AutM such that τ(r1) = r2.

Define the enumerator v(D,x) and the root polynomials r(M,x), r(M(D), x) as follows:

v(D,x) =∑

i≥1

ivixi; r(M,x) =

∑

i≥1

r(M, i)xi,

where r(M, i) denotes the number of non-transitive i-roots in M .

Theorem 2.12 enables us to get the following results by applying the enumerator and root

polynomial of M .

Theorem 2.13(Mao and Liu, [21]) The number rO(Γ) of non-isomorphic rooted maps on ori-

entable surfaces underlying a simple graph Γ is

rO(Γ) =

2ε(Γ)∏

v∈V (Γ)

(ρ(v) − 1)!

|AutΓ| ,

where ε(Γ),ρ(v) denote the size of Γ and the valency of the vertex v, respectively.

Theorem 2.14(Mao and Liu, [22]) The number rN (Γ) of rooted maps on non-orientable surfaces

underlying a graph Γ is

rN (Γ) =

(2β(Γ)+1 − 2)ε(Γ)∏

v∈V (Γ)

(ρ(v) − 1)!

∣∣∣Aut 12Γ∣∣∣

.

For a few well-known graphs, Theorems 2.13 and 2.14 enables us to get Table 2.

G rO(G) rN (G)

Pn n− 1 0

Cn 1 1

Kn (n− 2)!n−1 (2(n−1)(n−2)

2 − 1)(n− 2)!n−1

Km,n(m 6= n) 2(m− 1)!n−1(n− 1)!m−1 (2mn−m−n+2 − 2)(m− 1)!n−1(n− 1)!m−1

Kn,n (n− 1)!2n−2 (2n2−2n+2 − 1)(n− 1)!2n−2

Bn(2n)!2nn! (2n+1 − 1) (2n)!

2nn!

Dpn (n− 1)! (2n − 1)(n− 1)!

Dpk,ln (k 6= l) (n+k+l)(n+2k−1)!(n+2l−1)!

2k+l−1n!k!l!(2n+k+l−1)(n+k+l)(n+2k−1)!(n+2l−1)!

2k+l−1n!k!l!

Dpk,kn

(n+2k)(n+2k−1)!2

22kn!k!2(2n+2k−1)(n+2k)(n+2k−1)!2

22kn!k!2

Table 2

Apply the Burnside Lemma in permutation groups, we got the numbers of unrooted maps

of complete graph Kn on orientable or non-orientable surfaces by calculating the stabilizer of

each automorphism of complete maps.


Theorem 2.15(Mao, Liu and Tian, [23]) The number nO((Kn) of complete maps of order

n ≥ 5 on orientable surfaces is

nO(Kn) =1

2(∑

k|n+

∑

k|n,k≡0(mod2)

)(n− 2)!

nk

knk (n

k)!

+∑

k|(n−1),k 6=1

φ(k)(n − 2)!n−1

k

n− 1.

and n(K4) = 3.

Theorem 2.16(Mao, Liu and Tian, [23]) The number nN (Kn) of complete maps of order

n, n ≥ 5 on non-orientable surfaces is

nN (Kn) =1

2(∑

k|n+

∑

k|n,k≡0(mod2)

)(2α(n,k) − 1)(n− 2)!

nk

knk (n

k)!

+∑

k|(n−1),k 6=1

φ(k)(2β(n,k) − 1)(n− 2)!n−1

k

n− 1,

and nN (K4) = 8, where,

α(n, k) =

n(n − 3)

2k, if k ≡ 1(mod2);

n(n − 2)

2k, if k ≡ 0(mod2),

β(n, k) =

(n − 1)(n − 2)

2k, if k ≡ 1(mod2);

(n − 1)(n − 3)

2k, if k ≡ 0(mod2).

§3. Complex Networks with Reality

A network is a directed graph G associated with a non-negative integer-valued function c on

edges and conserved at each vertex, which are abstracting of practical networks, for instance,

the electricity, communication and transportation networks such as those shown in Fig.11 for

the high-speed rail network in China planed a few years ago.

Fig.11

Clearly, a network is nothing else but a labeled graph GL with L : E(G) → Z+. generally,

a labeled graph on a graph G = (V,E; I) is a mapping θL : V ∪E → L for a label set L, denoted

by GL. If θL : E → ∅ or θL : V → ∅, then GL is called a vertex labeled graph or an edge labeled

12 Linfan MAO

graph, denoted by GV or GE , respectively. Otherwise, it is called a vertex-edge labeled graph.

Similarly, two networks GL11 , GL2

2 are equivalent, if there is an isomorphism φ : G1 → G2 such

that φ(L1(x)) = L2(φ(x)) for x ∈ V (G1)⋃E(G2).

It should be noted that labeled graphs are more useful in understanding matters in the

universe. For example, there is a famous story, i.e., the blind men with an elephant. In this

story, 6 blind men were be asked to determine what is an elephant looks like. The man touched

the elephant’s leg, tail, trunk, ear, belly or tusk respectively claims it’s like a pillar, a rope,

a tree branch, a hand fan, a wall or a solid pipe. Each of them insisted on his own and not

accepted others.

Fig.12

They then entered into an endless argument. All of you are right! A wise man explained to

them: why are you telling it differently is because each one of you touched the different part

of the elephant. What is the meaning of the wise man? He claimed nothing else but the looks

like of an elephant, i.e.,

An elephant = {4 pillars}⋃

{1 rope}⋃

{1 tree branch}⋃ {2 hand fans}

⋃{1 wall}

⋃{1 solid pipe}.

Usually, a thing T is identified with known characters on it at one time, and this process is

advanced gradually by ours. For example, let µ1, µ2, · · · , µn be the known and νi, i ≥ 1 the

unknown characters at time t. Then, the thing T is understood by

T =

(n⋃

i=1

{µi})⋃⋃

k≥1

{νk}

in logic and with an approximation T ◦ =n⋃

i=1

{µi} at time t, which are both Smarandache

multispace ([7],[26]).

What is the implications of this story for understanding matters in the universe? It lies

in the situation that humans knowing matters in the universe is analogous to these blind men.

However, if the wise man were L.Euler, a mathematician he would tell these blind men that an

elephant looks like nothing else but a tree labeled by sets as shown in Fig.13, where, {a} =tusk,

{b1, b2} =ears, {c} =head, {d} =belly, {e1, e2, e3, e4} =legs and {f} =tail with their intersection


sets labeled on edges.l lll l ll ll la c d f

b1

b2

e1

e2

e3

e4

Fig.13

For the case of Euclidean space with dimension≥ 3, the intuition tells us that to embed

a graph in a of dimension is not difficult by the result following. However, it is obvious but a

universal skeleton inherited in all matters.

Theorem 3.1 A simple graph G can be rectilinear embedded, i.e., all edge are segment of

straight line in a Euclidean space Rn with n ≥ 3.

In fact, we can choose n district points in curve (t, t2, t3) of Euclidean space R3 on n different

values of t. Then, it can be easily show that all these straight lines are never intersecting.

Whence, it is a trivial problem on embedding graphs of R3. However, all matters are in 3-

dimensional Euclidean space in the eyes of humans, i.e., the reality of a matter in the universe

should be understanding on its 1-dimensional skeleton in the space.

Then, what is the reality of a matter? The word reality of a matter T is its state as it actually

exist, including everything that is and has been, no matter it is observable or comprehensible

by humans. How can we hold on the reality of matters? Usually, a matter T is multilateral,

i.e., Smarandache multispace or complex one and so, hold on its reality is difficult for humans

in logic, such as the meaning in the story of the blind men with an elephant.

For hold on the reality of matters, a general notion is

MatterDecompose−→ Microcosmic Particles

Abstract−→ Complex Network.

For example, the physics determine the nature of matters by subdividing a matter to an irre-

ducibly smallest detectable particle ([28]), i.e., elementary particles, which is essentially transfer

the matter to a complex network such as those meson’s and baryon’s composition by quarks.

Similarly, the basic unit of life or the basic unit of heredity are cells and genes in biology

which also enables us to get the life networks of cell or genes. This notion can be found in all

modern science with an conclusion that a matter = a complex network. Its essence of this notion

is to determine the nature of irreducibly smallest detectable units and then, holds on reality

of the matter. However, a matter can be always divided into submatters, then sub-submatters

and so on. A natural question on this notion is whether it has a terminal point or not. On the

other hand, it is a very large complex network in general. For example, the complex network of

a human body consists of 5×1014−−6×1014 cells. Are we really need such a large and complex

network for the reality of matters? Certainly not! How can we hold on the reality of matters

14 Linfan MAO

by such a complex network? And do we have mathematical theory on complex network? The

answer is not certain because although we have established a theory on complex network but

it is only a local theory by combination of the graphs and statistics with the help of computer

([3]), can not be used for the reality of matters.

However, we find a beacon light inspired by the traditional Chinese medicine. There are

12 meridians which completely reflects the physical condition of human body in traditional

Chinese medicine: LU, LI, ST, SP, HT, SI, BL, KI, PC, SJ, GB, LR. For example, the LI and

GB meridians are shown in Fig.14.

Fig.14

All of these 12 meridians can be classified into 3 classes following:

Class 1. Paths, including LU, LI, SP, HT, SI, KI, PC and LR meridians;

Class 2. Trees, including GB, ST and SJ meridians;

Class 3. Gluing Product of circuit with paths Cn

⊙Pm1

⊙Pm2 , including BL meridian.

According to the Standard China National Standard (GB 12346-90), the inherited graph

of the 12 meridians on a human body is shown in Fig.15.q qq qq

qq q qqq

qqqq qq

qST1

ST8

ST5

ST45

BL10

GB15

BL40

GB44BL67

q q qqqqq qqq q qKI27 LU1

SP6

LR1 SP1KI1

qqHT1

HT9

LR14PC1

SP21

q q qqq qPC9

LU11

BL1

qq q qq

q qqSJ23

SJ20 SJ16

SJ1

qq q qq

GB1

SI18

SI1

SI19qq qqq q qq qqq q

q qqqqq

�� qq qq qqqqq q qq q

qqqq qqq qq qqq qqqq

qq qqq

qFig.15 12 Meridian graph on a human body


By the traditional Chinese medicine ([28]), if there exists an imbalanced acupoint on one of

the 12 meridians, this person must has illness and in turn, there must be imbalance acupoints

on the 12 meridians for a patient. Thus, finding out which acupoint on which meridian is in

imbalance with Yin more than Yang or Yang more than Yin is the main duty of a Chinese

doctor. Then, the doctor regulates the meridian by acupuncture or drugs so that the balance

on the imbalance acupoints recovers again, and then the patient recovers.

Then, what is the significance of the treatment theory in traditional Chinese medicine

to science? It implies we are not need a large complex network for holding on the body of

human. Whether or not classically mathematical elements enough for understanding complex

networks, i.e., matters in the universe? The answer is negative because all of them are local.

Then, could we establish a mathematics over elements underlying combinatorial structures? The

answer is affirmative, i.e., mathematical combinatorics discussed in this paper. Certainly, we

can introduce 2 kinds of elements respectively on living of non-living matters.

Element 1(Non-Living Body). A continuity flow−→GL is an oriented embedded graph−→

G in a topological space S associated with a mapping L : v → L(v), (v, u) → L(v, u), 2

end-operators A+vu : L(v, u) → LA+

vu(v, u) and A+uv : L(u, v) → LA+

uv (u, v) on a Banach space B

over a field F such as those shown in Fig.16,-�� L(v, u)A+vu A+

uvL(v) L(u)

v uFig.16

with L(v, u) = −L(u, v), A+vu(−L(v, u)) = −LA+

vu(v, u) for ∀(v, u) ∈ E(−→G)

and holding with

continuity equation ∑

u∈NG(v)

LA+vu (v, u) = L(v) for ∀v ∈ V

(−→G).

Element 2(Living Body). A harmonic flow−→GL is an oriented embedded graph

−→G in a

topological space S associated with a mapping L : v → L(v) − iL(v) for v ∈ E(−→G)

and L :

(v, u) → L(v, u)− iL(v, u), 2 end-operators A+vu : L(v, u)− iL(v, u) → LA+

vu(v, u)− iLA+vu(v, u)

and A+uv : L(v, u) − iL(v, u) → LA+

uv(v, u) − iLA+uv(v, u) on a Banach space B over a field F

such as those shown in Fig.17, -�� L(v, u) − iL(v, u)A+vu A+

uvL(v) L(u)

v uFig.17

where i2 = −1, L(v, u) = −L(u, v) for ∀(v, u) ∈ E(−→G)

and holding with continuity equation

∑

u∈NG(v)

(LA+

vu (v, u) − iLA+vu (v, u)

)= L(v) − iL(v)

for ∀v ∈ V(−→G).

16 Linfan MAO

Let G be a closed family of graphs−→G under the union operation and let B be a linear

space (B; +, ·), or furthermore, a commutative ring, a Banach or Hilbert space (B; +, ·) over a

field F . Denoted by (GB; +, ·) and(G

±B

; +, ·)

the respectively elements 1 and 2 form by graphs

G ∈ G . Then, elements 1 and 2 can be viewed as vectors underlying an embedded graph G in

space, which enable us to establish mathematics on complex networks and get results following.

Theorem 3.2([9-10,14-18]) If G is a closed family of graphs−→G under the union operation and

B a linear space (B; +, ·), then, (GB; +, ·) and(G

±B

; +, ·)

with linear operators A+vu, A+

uv for

∀(v, u) ∈ E

( ⋃G∈G

−→G

)under operations + and · form respectively a linear space, and further-

more, a commutative ring if B is a commutative ring (B; +, ·) over a field F .

Theorem 3.3([9-10,14-18]) If G is a closed family of graphs under the union operation and

B a Banach space (B; +, ·), then, (GB; +, ·) and(G

±B

; +, ·)

with linear operators A+vu, A+

uv for

∀(v, u) ∈ E

( ⋃G∈G

−→G

)under operations + and · form respectively a Banach or Hilbert space

respect to that B is a Banach or Hilbert space.

A few well-known results such as those of Banach theorem, closed graph theorem and

Hahn-Banach theorem are also generalized on elements 1 and 2. For example, we obtained

results following.

Theorem 3.4(Taylor, [15]) Let−→GL ∈

⟨−→G i, 1 ≤ i ≤ n

⟩R×Rn

and there exist kth order derivative

of L to t on a domain D ⊂ R, where k ≥ 1. If A+vu, A+

uv are linear for ∀(v, u) ∈ E(−→G), then

−→GL =

−→GL(t0) +

t− t01!

−→GL′(t0) + · · · + (t− t0)

k

k!

−→GL(k)(t0) + o

((t− t0)

−k −→G),

for ∀t0 ∈ D , where o((t− t0)

−k −→G)

denotes such an infinitesimal term L of L that

limt→t0

L(v, u)

(t− t0)k

= 0 for ∀(v, u) ∈ E(−→G).

Particularly, if L(v, u) = f(t)cvu, where cvu is a constant, denoted by f(t)−→GLC with LC :

(v, u) → cvu for ∀(v, u) ∈ E(−→G)

and

f(t) = f(t0) +(t − t0)

1!f′(t0) +

(t − t0)2

2!f′′(t0) + · · · + (t − t0)

k

k!f

(k)(t0) + o((t − t0)

k)

,

then

f(t)−→GLC = f(t) · −→GLC .

Theorem 3.5(Hahn-Banach, [19]) Let H±

Bbe an element 2 subspace of G

±B

and let F : H±

B→

C be a linear continuous functional on H±

B. Then, there is a linear continuous functional

F : G±B

→ C hold with


(1) F(−→GL2

)= F

(−→GL2

)if−→GL2 ∈ H

±B

;

(2)∥∥∥F∥∥∥ = ‖F‖.

For applications of elements 1 and 2 to physics and other sciences such as those of ele-

mentary particles, gravitations, ecological system, · · · etc., the reader is refereed to references

[11]-[13] and [18]-[20] for details.

References

[1] J.A.Bondy and U.S.R.Murty, Graph Theory with Applications, The Macmillan Press Ltd,

1976.

[2] G.Chartrand and L.Lesniak, Graphs & Digraphs, Wadsworth, Inc., California, 1986.

[3] G.R.Chen, X.F.Wang and X.Li, Introduction to Complex Networks – Models, Structures

and Dynamics (2 Edition), Higher Education Press, Beijing, 2015.

[4] J.L.Gross and T.W.Tucker, Topological Graph Theory, John Wiley & Sons,1987.

[5] Y.P.Liu, Enumerative Theory of Maps, Kluwer Academic Publishers , Dordrecht/ Boston/

London,1999.

[6] Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries (Sec-

ond edition), The Education Publisher Inc., USA, 2011.

[7] Linfan Mao, Smarandache Multi-Space Theory, The Education Publisher Inc., USA, 2011.

[8] Linfan Mao, Combinatorial Geometry with Applications to Field Theory, The Education

Publisher Inc., USA, 2011.

[9] Linfan Mao, Mathematics on non-mathematics - A combinatorial contribution, Interna-

tional J.Math. Combin., Vol.3(2014), 1-34.

[10] Linfan Mao, Extended Banach−→G -flow spaces on differential equations with applications,

Electronic J.Mathematical Analysis and Applications, Vol.3, No.2 (2015), 59-91.

[11] Linfan Mao, A new understanding of particles by−→G -flow interpretation of differential

equation, Progress in Physics, Vol.11(2015), 193-201.

[12] Linfan Mao, A review on natural reality with physical equation, Progress in Physics,

Vol.11(2015), 276-282.

[13] Linfan Mao, Mathematics with natural reality – Action Flows, Bull.Cal.Math.Soc., Vol.107,

6(2015), 443-474.

[14] Linfan Mao, Mathematical combinatorics with natural reality, International J.Math. Com-

bin., Vol.2(2017), 11-33.

[15] Linfan Mao, Hilbert flow spaces with operators over topological graphs, International

J.Math. Combin., Vol.4(2017), 19-45.

[16] Linfan Mao, Complex system with flows and synchronization, Bull.Cal.Math.Soc., Vol.109,

6(2017), 461-484.

[17] Linfan Mao, Mathematical 4th crisis: to reality, International J.Math. Combin., Vol.3(2018),

147-158.

[18] Linfan Mao, Harmonic flow’s dynamics on animals in microscopic level with balance re-

covery, International J.Math. Combin., Vol.1(2019), 1-44.

18 Linfan MAO

[19] Linfan Mao, Science’s dilemma – A review on science with applications, Progress in Physics,

Vol.15(2019), 78–85.

[20] Linfan Mao, A new understanding on the asymmetry of matter-antimatter, Progress in

Physics, Vol.15, 3(2019), 78-85.

[21] Linfan Mao and Yanpei Liu, On the roots on orientable embeddings of graph, Acta Math.

Scientia, 23A(3), 287-293(2003).

[22] Linfan Mao and Yanpei Liu, Group action approach for enumerating maps on surfaces, J.

Applied Math. & Computing, Vol.13, No.1-2,201-215.

[23] Linfan Mao,Yanpei Liu and Feng Tian, The number of complete maps on surfaces, arXiv:

math.GM/0607790. Also in L.Mao ed., Selected Papers on Mathematical Combinatorics(I),

World Academic Union, 2006, 94-118.

[24] W.S.Massey, Algebraic topology: An introduction, Springer-Verlag, New York, etc., 1977.

[25] Quang Ho-Kim and Pham Xuan Yem, Elementary Particles and Their Interactions, Springer-

Verlag Berlin Heidelberg, 1998.

[26] F.Smarandache, Paradoxist Geometry, State Archives from Valcea, Rm. Valcea, Romania,

1969, and in Paradoxist Mathematics, Collected Papers (Vol. II), Kishinev University

Press, Kishinev, 5-28, 1997.

[27] W.T.Tutte, What is a map? in New Directions in the Theory of Graphs, ed. by F.Harary,

Academic Press, 1973, 309-325.

[28] Zhichong Zhang, Comments on the Inner Canon of Emperor(Qing Dynasty, in Chinese),

Northern Literature and Art Publishing House, 2007.


Some Subclasses of

Meromorphic with p-Valent q-Spirallike Functions

Read.S.A.Qahtan, Hamid Shamsan and S.Latha

(Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore 570005, India)

E-mail: [email protected], [email protected], [email protected]

Abstract: In this paper, we introduce and investigate two new subclasses of the function

class ΣMS(p, λ, β, q) and ΣMC(p, λ, β, q) of q-spirallike meromorphic functions defined in

the punctured open unit disc.

Key Words: Univalent functions, meromorphic functions, meromorphic q-spirallike func-

tions.

AMS(2010): 30C45.

§1. Introduction

Let Σp be the class of functions f of the form

f(z) =1

zp+

∞∑

n=1−p

an zn, (p ∈ N = 1, 2, 3, · · · ), (1)

which are analytic in the open disc E∗ = {z : z ∈ C and 0 < |z| < 1}. Let S be the subclass of

functions in Σp which are univalent in E. Let P be the class of functions p given by

p(z) = 1 +

∞∑

n=1

pn zn (z ∈ E), (2)

which are analytic in the open disc E and satisfy the condition:

ℜ{p(z)} > 0 (z ∈ E). (3)

If f ∈ Σp and satisfies

−ℜ{zf ′(z)

f(z)

}> β (z ∈ E, 0 ≤ β < p), (4)

then we say that f is meromorphic p-valent starlike of order β (0 ≤ β < p) and we denote this

class by ΣMS(p, β).

1Received July 11, 2019, Accepted November 22, 2019.

20 Read.S.A.Qahtan, Hamid Shamsan and S.Latha

If f ∈ Σp and satisfies

−ℜ{

1 +zf ′′(z)

f ′(z)

}> β (z ∈ E, 0 ≤ β < p), (5)

then we say that f is meromorphic p-valent convex of order β and we denote this class by

f ∈ ΣMC(p, β).

A function f ∈ Σp is said to be λ-spirallike of order β in the unit disk E if

−ℜ{eiλ zf

′(z)

f(z)

}> β, (z ∈ E, 0 ≤ β < p, |λ| < π

2).

In [8] Jackson introduced and studied the concept of the q-derivative operator ∂q as

follows

∂qf(z) =f(z) − f(qz)

z (1 − q), (z 6= 0, 0 < q < 1, ∂qf(0) = f ′(0)). (6)

Equivalently (6) may be written as

∂qf(z) = 1 +

∞∑

n=2

[n]q an zn−1, z 6= 0, (7)

where [n]q = 1− qn

1− q. Note that as q → 1−, [n]q → n.

Definition 1.1 A function f ∈ Σp is said to be meromorphic p-valent λ-q-spirallike functions

of order β if it satisfies the following:

−ℜ{eiλ z∂qf(z)

f(z)

}> β (z ∈ E, |λ| < π

2, 0 ≤ β > p cosλ, 0 < q ≤ 1), (8)

we denote this class by ΣMS(p, λ, β, q).

Definition 1.2 A function f ∈ Σp is said to be meromorphic p-valent convex λ-q-spirallike

functions of order β if it satisfies the following

−ℜ{eiλ ∂q(z∂qf(z))

∂qf(z)

}> β (z ∈ E, 0 ≤ β < 1), (9)

we denote this class by ΣMC(p, λ, β, q).

Remark 1.1 f ∈ MS(p, λ, β, q) iff

−eiλ z∂qf(z)

f(z)≺ peiλ − (2β − pe−iλ)z

1 − z, (10)

and f ∈ MC(p, λ, β, q) iff

−eiλ

(∂q(z∂qf(z))

∂qf(z)

)≺ peiλ − (2β − pe−iλ)z

1 − z. (11)

Some Subclasses of Meromorphic with p-Valent q-Spirallike Functions 21

§2. Main Results

Theorem 2.1 If the sequence {Ap+m}∞0 defined by

Ap = 2(β−p cos λ)p+[p]q

, m = 0,

Ap+m = 2(β−p cos λ)p+[p+m]q

(1 +

∑m−1k=0 |ap+k|

), m ∈ N ,

(12)

and p ∈ N. Then

Ap+m =2(β − p cosλ)

2β + [m+ p]q + p− 2p cosλ

m∏

k=0

2β + [k + p]q + p− 2p cosλ

p+ [p+ k]q, (13)

where m ∈ N0 = N \ {0}.

Proof By virtue of (12), we have

p+ [p+m+ 1]qAp+m+1 = 2(β − p cosλ)

(1 +

m∑

k=0

Ap+k

), (14)

and

p+ [p+m]qAp+m = 2(β − p cosλ)

(1 +

m−1∑

k=0

Ap+k

). (15)

From (14) and (15), we have

Ap+m+1

Ap+m

=2β + [m+ p]q + p− 2p cosλ

p+ [p+m+ 1]qm ∈ N0. (16)

Ap+m =Ap+m

Ap+m−1.Ap+m−1

Ap+m−2· · · Ap+1

Ap

.Ap

=2β + [m+ p− 1]q + p− 2p cosλ

p+ [p+m]q· · · 2β + [p]q + p− 2p cosλ

p+ [p+ 1]q.2β − 2p cosλ

p+ [p]q

=2(β − p cosλ)


m∏

k=0

2β + [k + p]q + p− 2p cosλ

p+ [p+ k]q(m ∈ N).

(17)

The proof of Theorem 2.1 is completed. 2As q → 1−, we get the following result proved by Shi.et al. [13].

Corollary 2.1 If {Ap+m}∞0 defined by

Ap = β−p cos λ

p, m = 0,

Ap+m = 2(β−p cos λ)2p+m

(1 +

∑m−1k=0 |ap+k|

), m ∈ N ,

(18)


and p ∈ N. Then

Ap+m ≤ 2(β − p cosλ)

2β +m+ 2p− 2p cosλ

m∏

k=0

2β + k + 2p− 2p cosλ

2p+ k, (19)

where, m ∈ N0 = N \ {0}.

Theorem 2.2 Let f(z) = 1zp +

∑∞m=0 ap+mz

p+m ∈ MS(p, λ, β, q). Then

|ap+m| ≤ 2(β − p cosλ)


m∏

k=0

2β + [k + p]q + p− 2p cosλ

p+ [p+ k]q(m ∈ N0). (20)

Proof Let

L(z) =β + eiλ z∂qf(z)

f(z) + ip sinλ

β − p cosλ(z ∈ E, f ∈ MS(p, λ, β, q)). (21)

We know that L ∈ P . It follows that

eiλz∂qf(z) = (β − p cosλ)f(z)L(z) − (β − ip sinλ)f(z). (22)

Let

L(z) = 1 + l1z + l2z2 + · · · . (23)

Then

eiβ

(−[p]qzp

+ [p]qapzp + [p+ 1]qap+1z

p+1 + · · · + [p+m]qap+mzp+m + · · ·

)

= (β − p cosλ)

(1

zp+ apz

p + ap+1zp+1 + · · ·

)× (1 + l1z + l2z

2 + · · · )

− (β − ip sinλ)

(1

zp+ apz

p + ap+1zp+1 + · · · + ap+mz

p+m + · · ·).

(24)

We have from sides (24)

eiλ[p+m]qap+m = (β − p cosλ) (l2p+m + aplm + ap+mlm−1

+ · · · +ap+m) − (β + ip sinλ)ap+m. (25)

Moreover, we know that

|ln| ≤ 2 (n ∈ N). (26)

From (25) and(26) we have

|ap| ≤2(β − p cosλ)

p+ [p]q(27)


and

|ap+m| ≤ 2(β − p cosλ)

p+ [p+m]q

(1 +

m−1∑

k=0

|ap+k|)

(28)

with supposing p ∈ N . We define {Ap+m}∞m=0 by

Ap = 2(β−p cos λ)

p+[p]q, m = 0;

Ap+m = 2(β−p cos λ)p+[p+m]q

(1 +

∑m−1k=0 |ap+k|

), m ≥ 1.

(29)

Now by the mathematical induction principle we will prove that

|ap+m| ≤ Ap+m(m ∈ N0). (30)

We can easily verify that

|ap| ≤ Ap =2(β − p cosλ)

p+ [p]q. (31)

Thus, assuming that

|ap+j | ≤ Ap+j(j = 0, 1, · · · ,m.m ∈ N0), (32)

from (28)) and (32) we have

|ap+m+1| ≤2(β − p cosλ)

p+ [p+m+ 1]q

(1 +

m∑

k=0

|ap+k|)

≤ 2(β − p cosλ)

p+ [p+m+ 1]q

(1 +

m∑

k=0

|Ap+k|)

=Ap+m+1 (m ∈ N0).

(33)

Therefore, by the principle of mathematical induction, we have

|ap+m| ≤ Ap+m (m ≤ N0). (34)

By means of Theorem 2.1 and (29), we know that



m∏

k=0

2β + [k + p]q + p− 2p cosλ

p+ [p+ k]q(m ∈ n0). (35)

So, from (34) and (35) we get proof of the Theorem 2.2. 2As q → 1− we get the following result proved by Shi.et al. [13].

Corollary 2.2 Let f(z) = 1zp +

∑∞m=0 ap+mz

p+m ∈ MS(p, λ, β). Then


2β +m+ 2p− 2p cosλ

m∏

k=0

2β + k + 2p− 2p cosλ

2p+ k(m ∈ n0). (36)


From Theorem 2.2 we get the following result.


∑∞m=0 ap+mz

p+m ∈ MC(p, λ, β, q). Then

Ap+m =2p(β − p cosλ)

[p+m](2β + [m+ p]q + p− 2p cosλ)

m∏

k=0

2β + [k + p]q + p− 2p cosλ

p+ [p+ k]q(m ∈ n0). (37)

Theorem 2.3 Let f(z) = 1zp +

∑∞m=0 ap+mz

p+m ∈ MS(p, λ, β, q). Then

p cosλ− 2(β − p cosλ)r

1 − r≤ ℜ

(−eiϑ z∂qf(z)

f(z)

)≤ p cosλ+ 2(β − p cosλ)r

1 + r(38)

for |z| = r < 1.

Proof Suppose the function φ defined by

φ(z) =peiλ − (2β − pe−iλz)

1 − z(z ∈ E). (39)

Let z = reiλ (0 < r < 1). We have

ℜ{φ(z)} = p cosλ− 2(β − p cosλ)r(cos ϑ− r)

1 + r2 − 2r cosϑ. (40)

Let

ϕ(τ) = p cosλ− 2(β − p cosλ)r(τ − r)

1 + r2 − 2τr(τ = cosϑ). (41)

Then

∂qϕ(τ) =−2r(β − p cosλ)[(1 + r2 − 2τr) − r[2r]q(τ − r)]

(1 + r2 − 2qrτ)(1 + r2 − 2rτ). (42)

This means that


1 − r≤ ℜ(φ(z)) ≤ p cosλ+

2(β − p cosλ)r

1 + r, (43)

which is equivalent to


1 − r≤ ℜ{φ(z)} ≤ p cosλ+ 2(β − p cosλ)r

1 + r. (44)

We know that

−eiλ z∂qf(z)

f(z)≺ φ(z)

and φ(z) is univalent in E, this is prove the inequality (38). 2As q → 1− we get the following result proved by Shi.et al. [13].



∑∞m=0 ap+mz

p+m ∈ MS(p, λ, β). Then


1 − r≤ ℜ

(−eiλ zf

′(z)

f(z)


1 + r, (45)

for |z| = r < 1.


∑∞m=0 ap+mz

p+m ∈ MC(p, λ, β, q). Then


1 − r≤ ℜ

(−eiλ ∂q(z∂qf(z))

∂qf(z)


1 + r(46)

for |z| = r < 1.

Theorem 2.4 If f ∈ Σp satisfies

q∞∑

n=1−p

(|[n]qe

iλ + γ| + |[n]qeiλ + 2β − γ|

)|an| ≤ |[p]qeiλ − 2qβ + qγ| − |[p]qeiλ − qγ| (47)

for some real λ, β and γ (0 ≤ γ ≤ p cosλ), then f ∈ MS(p, λ, β, q)

Proof To prove f ∈ MS(p, λ, β, q), it suffices to show that

∣∣∣∣∣∣

eiλ z∂qf(z)f(z) + γ

eiλ z∂qf(z)f(z) + (2β − γ)

∣∣∣∣∣∣< 1 (z ∈ E, 0 ≤ γ ≤ p cosλ). (48)

From (47), we know that

∣∣[p]qeiλ − 2qβ + qγ∣∣+ q

∞∑

n=1−p

∣∣[n]qeiλ + 2β − γ

∣∣ |an| ≥∣∣[p]qeiλ − qγ

∣∣

+q

∞∑

n=1−p

|[n]qeiλ + γ||an| > 0. (49)

Now, by the maximum modulus principle, we deduce from (1) and (49) that

∣∣∣∣∣∣

eiλ z∂qf(z)f(z) + γ

eiλ z∂qf(z)f(z) + (2β − γ)

∣∣∣∣∣∣=

∣∣∣∣∣∣

eiλ(

−[p]qqzp +

∑∞n=1−p[n]qanz

n)

+ γzp + γ

∑∞n=1−p anz

n

eiλ

(−[p]qqzp +

∑∞n=1−p[n]qanzn

)+ (2β − γ)

(1zp +

∑∞n=1−p anzn

)

∣∣∣∣∣∣

=

∣∣∣∣∣(−[p]qe

iλ + qγ) + q∑∞

n=1−p([n]qeiλ + γ)anz

n

(−[p]qeiλ + 2qβ − qγ) + q∑∞

n=1−p([n]qeiλ + 2β − γ)anzn

∣∣∣∣∣

<|[p]qeiλ − qγ| + q

∑∞n=1−p |[n]qe

iλ + γ||an||[p]qeiλ − 2qβ + qγ| − q

∑∞n=1−p |[n]qeiλ + 2β − γ||an|

≤1.

(50)

This completes the proof. 2


As q → 1− we get the following result proved by Shi.et al.[13].

Corollary 2.6 If f ∈ Σp satisfies the

∞∑

n=1−p

(|neiλ + γ| + |neiλ + 2β − γ|

)|an| ≤ |peiλ − 2β + γ| − |peiλ − γ| (51)

for some real λ, β and γ (0 ≤ γ ≤ p cosλ), then f ∈ MC(p, λ, β).

Corollary 2.7 If f ∈∑@p satisfies the

q

∞∑

n=1−p

|[n]q|(|[n]qe

iλ + γ| + |[n]qeiλ + 2β − γ|

)|an| ≤ [p]q

(|[p]qeiλ − 2qβ + qγ|

− |[p]qeiλ − qγ|)

(52)

for some real λ, β and γ (0 ≤ γ ≤ p cosλ), then f ∈ MC(p, λ, β, q).

Lemma 2.1([7]) If |φ| attains its maximum value on the circle |z| = r < 1 at z0 and φ is a

nonconstant regular function in E then

z0φ′(z0) = kφ(z0), k ≥ 1, k ∈ R.

Theorem 2.5 If f ∈ Σp satisfies

∣∣∣∣∣f(z)

f(qz)+zf(z)∂2

qf(z)

f(qz)∂qf(z)− z∂qf(z)

f(qz)

∣∣∣∣∣ <β − p

2β(53)

for some real β > p, than f ∈ MS(p, 0, β, q).

Proof Define the function ϕ by

ϕ(z) =

z∂qf(z)f(z) + p

z∂qf(z)f(z) + (2β − p)

(z ∈ E). (54)

Note that ϕ is analytic in E and ϕ(0) = 0. From (54), we have

z∂qf(z)

f(z)=

−p+ (2β − p)ϕ(z)

1 − ϕ(z). (55)

Taking q-differentiating of (55) logarithmically, we get

f(z)

f(qz)+zf(z)∂2

qf(z)


f(qz)=

z(1 − ϕ(z))(2β − p)∂qϕ(z)

(−p+ (2β − p)ϕ(z))(1 − ϕ(qz))+

z∂qϕ(z)

(1 − ϕ(qz)). (56)


From (53) and (56), we get that

∣∣∣∣∣f(z)

f(qz)+zf(z)∂2

qf(z)


f(qz)

∣∣∣∣∣ =

∣∣∣∣2(β − p)∂qϕ(z)

[−p+ (2β − p)ϕ(z)](1 − ϕ(qz))

∣∣∣∣ <β − p

2β. (57)

Consider z0 ∈ E such that

max|z|≤|z0|

|ϕ(z)| = |ϕ(z0)| = 1.

By Lemma 2.1, let ϕ(z0) = eiϑ and z0∂qϕ(z0) = Leiϑ (L ≥ 1). For such a point z0, we have

that ∣∣∣∣∣f(z0)

f(qz0)+z0f(z0)∂

2qf(z0)

f(qz0)∂qf(z0)− z0∂qf(z0)

f(qz0)

∣∣∣∣∣

=

∣∣∣∣2(β − p)Leiϑ

[−p+ (2β − p)eiϑ](1 − eiϑ)

∣∣∣∣

=2(β − p)L√

p2 + (2β − p)2 − 2p(2β − p) cosϑ√

2(1 − cosϑ)

≥ β − p

2β.

(58)

This contradicts our condition (53). Therefore ,there is no z0 ∈ E such that |ϕ(z0)| = 1. This

implies that |ϕ(z)| < 1 (z ∈ E∗), that is,

∣∣∣∣∣∣

z∂qf(z)f(z) + p

z∂qf(z)f(z) + (2β − p)

∣∣∣∣∣∣< 1, (z ∈ E)

thus, we conclude that f ∈ MS(p, 0, β, q). 2As q → 1− we get the following result proved by Shi.et al. [13].

Corollary 2.8 If f ∈ Σp satisfies

∣∣∣∣1 +zf ′′(z)

f ′(z)− zf ′(z)

f(z)

∣∣∣∣ <β − p

2β(59)

for some real β > p, than f ∈ MS(p, 0, β).

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On the Complexity of Some Classes of Circulant Graphs and

Chebyshev Polynomials

S. N. Daoud1,2, Muhammad Kamran Siddiqui3

1.Department of MathematicsFaculty of Science, Taibah University, Al-Madinah 41411, Saudi Arabia

2. Department of Mathematics and Computer ScienceFaculty of Science, Menoufia University, Shebin El Kom 32511, Egypt

3.Department of MathematicsCOMSATS University Islamabad, Lahore Campus, 54000, Pakistan

E-mail: [email protected], [email protected]

Abstract: Deriving closed formulae of the number of spanning trees for various graphs has

attracted the attention of a lot of researchers. In this paper we derive simple and explicit

formulas for the number of spanning trees in many classes of circulant graphs using the

properties of Chebyshev polynomials. Deriving closed formulae of the number of spanning

trees for various graphs has attracted the attention of a lot of researchers. In this paper

we derive simple and explicit formulas for the number of spanning trees in many classes of

circulant graphs using the properties of Chebyshev polynomials.

Key Words: Number of spanning trees, circulant graphs, Chebyshev polynomials.

AMS(2010): 05C78.

§1. Introduction

The number of spanning trees τ(G) in a graph G (networks) is an important invariant. We

call τ(G) the complexity of G. The evaluation of this number is not only interesting from a

mathematical (computational) perspective, but also, it is an important measure of reliability of

a network and designing electrical circuits. Some computationally hard problems such as the

travelling salesman problem can be solved approximately by using spanning trees. In this work

we consider finite undirected graph with no loops or multiple edges. Let G be such a graph of

n vertices. A spanning tree for a graph G is a subgraph of G that is a tree and contains all

vertices of τ(G). The number of spanning trees of G, is the total number of distinct spanning

subgraphs of G that are trees. A classic result of Kirchhoff [?] can be used to determine the

number τ(G) for G(V,E). Let V = {v1, v2, · · · , vn}. The Kirchhoff matrix H is defined as n×ncharacteristic matrix H = D−A, where D is the diagonal matrix of the degrees of G and A is

the adjacency matrix of G. Then the matrix H − [aij ] is defined as follows:

(1) aij , when vi and vj are adjacent and i 6= j;

(ii) aij , is equal to the degree of vertex vi if i = j;

(iii) aij = 0, otherwise. All of co-factors of H are equal to τ(G).

1Received April 25, 2019, Accepted November 23, 2019.

30 S. N. Daoud and Muhammad Kamran Siddiqui

There are more than one method for calculating τ(G). Let µ1 ≥ µ2 ≥ · · · ≥ µp denote the

eigenvalues of matrix of a p point graph. It can be easily shown that µp = 0. Kelmans and

Chelnokov [2] proved that The formula for the number of spanning trees in a d-regular graph

can be expressed as

τ(G) =1

p

[ p−1∏

k=1

(d− µk)]

where µ0 = d, µ1, µ2, · · · , µp−1 are the eigenvalues of the corresponding adjacency matrix of

the graph. Many works have conceived techniques to derive the number of spanning tree of a

graph can be found at [3-12]. The circulant graphs are an important class of graphs. Among

other applications, they are used in the design of local area networks, see [13-19].

Let 1 ≤ a1 ≤ a2 ≤ a3 ≤ · · · ≤ ak ≤ n2 , where n and ai(i = 1, 2, . . . , k) are positive integers.

An undirected circulant graph Cn(a1, a2, a3, · · · , ak) is a regular graph whose set of vertices is

V = {0, 1, 2, [dots, n−1} and whose set of edges is E = {i, i+ai(mod n)/i = 0, 1, 2, · · · , n−1, j =

1, 2, · · · , k}. If ak ≤ n2 , then Cn(a1, a2, a3, · · · , ak) is a 2k-regular graph; if ak = n

2 , then it

is a 2k − 1-regular one, see Nikolopoulos [20] and Papadopoulos [21]. The well known formula

τ(Cn(1, 2)) = nF 2n , where Fn is the nth Fibonacci number, see Kleiman, and Golden [22]. We

have obtained another proof for this formula in Theorem 3.3. The formulas of τ(C2n(1, n)),

τ(C3n(1, n)), τ(C4n(1, n)) can be found in Yuanping, et. al.[23].

§2. Chebyshev Polynomial

In this section we introduce some relations concerning Chebyshev polynomials of the first and

second kind which we use it in our computations. We begin from their definitions, see Yuanping,

et. al.[24]. Let An(x) be n× n matrix such that:

An(x) =

2x −1 0 . . . . . .

−1 2x −1. . .

...

0. . .

. . .. . . 0

.... . .

. . . 2x −1...

. . . 0 −1 2x

where all other elements are zeros.

Further, we recall that the Chebyshev polynomials of the first kind are defined by:

Tn(x) = cos(n arccos). (1)

The Chebyshev polynomials of the second kind are defined by

Un−1(x) =1

n

d

dxTn(x) =

sin(n arccos)

sin(arccos). (2)

On the Complexity of Some Classes of Circulant Graphs and Chebyshev Polynomials 31

It is easily verified that

Un(x) − 2xUn−1(x) + Un−2(x) = 0. (3)

It can then be shown from this recursion that by expanding one gets

Un(x) = det(An(x)), n ≥ 1. (4)

Furthermore by using standard methods for solving the recursion (3), one obtains the

explicit formula

Un(x) =1

2√x2 − 1

[(x+

√x2 − 1

)n+1 −(x−

√x2 − 1

)n+1], n ≥ 1, (5)

where the identity is true for all complex (except at x = ±1 , where the function can be taken

as the limit). The definition of easily yields its zeros and it can therefore be verified that

Un−1(x) = 2n−1[ n−1∏

j=1

(x− cosjπ

n)]. (6)

One further notes that

Un−1(−x) = (−1)n−1Un−1(x). (7)

These two results yield another formula for Un(x)

U2n−1(x) = 4n−1

[ n−1∏

j=1

(x2 − cos2jπ

n)]. (8)

Finally, a simple manipulation of the above formula yields the following formula (9), which

is extremely useful to us latter:

U2n−1

(√x+ 2

4

)=[ n−1∏

j=1

(x− 2 cosjπ

n)]

(9)

Furthermore, one can show that

U2n−1(x) =

1

2(1 − x2)

[1 − T2n

]=

1

2(1 − x2)

[1 − T2n(2 − 2x2)

](10)

and

Tn(x) =1

2

[(x+

√x2 − 1

)n+(x−

√x2 − 1

)n]. (11)

§3. Main Results

In our main results, i.e., Theorems 3.1 - 3.6 we use the following conclusion.

Lemma 3.1([25]) The Kirchhoff matrix of the circulant graph Cn(s1, s2, s3, · · · , sk) has n


eigenvalues, namely: 0 and the value 2k−ε−s1j −· · ·−ε−skj −ε−s1j −· · ·−ε−skj with ε = e2πjn

for any j = {1, 2, · · · , n− 1}

Corollary 3.2 For the circulant graph Cn(s1, s2, s3, · · · , sk),

τ(Cn(s1, s2, s3, · · · , sk)) =1

n

n−1∏

j=1

(2k − ε−s1j − · · · − ε−skj − ε−s1j − · · · − ε−skj

)

=1

n

n−1∏

j=1

(k∑

i=1

(2 − 2 cosjsiπ

n)

) .

Proof The proof follows immediately from Lemma 3.1. 2Theorem 3.3 For the spanning trees of C12n with three jumps 1, 2n, 3n , we have:

τ(C12n(1, 2n, 3n)) =n

12

(√

7

4+

√3

4

)4n

+

(√7

4−√

3

4

)4n

− 1

2

× n

12

(√

7

4+

√3

4

)2n

+

(√7

4−√

3

4

)2n

+ 1

2

× n

12

(√

5

2+

√3

2

)2n

+

(√5

2−√

3

2

)2n

2

× n

12

[(√2 + 1

)n

+(√

2 − 1)n]2

× n

12

(√

11

4+

√7

4

)2n

+

(√11

4−√

7

4

)2n

− 1

2

Proof Let ε = e2πi12n . Applying Lemma 3.1, we have

τ(C12n(n, 2n, 3n)) =1

12n

12n−1∏

j=1

(6 − ε−j − ε−2nj − ε−3nj − εj − ε2nj − ε3nj

)

=1

12n

12n−1∏

j=1

(6 − 2 cos

2πj

12n− 2 cos

4πjn

12n− 2 cos

6πjn

12n

)

=1

12n

12n−1∏

j=1

(6 − 2 cos

2πj

12n− 2 cos

πj

3− 2 cos

πj

2

)


=1

12n

12n−1∏

j=1,2∤j

(6 − 2 cos

2πj

12n− 2 cos

πj

3

)

×12n−1∏

j=1,2|j

(6 − 2 cos

2πj

12n− 2 cos

πj

3− 2 cos

πj

2

)

=1

12n

12n−1∏

j=1

(6 − 2 cos

2πj

12n− 2 cos

πj

3

)

×12n−1∏

j=1

6 − 2 cos 2πj12n

− 2 cos πj3 − 2 cos πj

2

6 − 2 cos 2πj12n

− 2 cos πj3

If we put j = 2j′ in the second term for some integer j′ we get

τ(C12n(n, 2n, 3n)) =1

12n

12n−1∏

j=1

(6 − 2 cos

2πj

12n− 2 cos

πj

3

)

×6n−1∏

j=1

6 − 2 cos 2πj6n

− 2 cos 2πj3 − 2 cosπj

6 − 2 cos 2πj6n

− 2 cos πj3

=1

12n

12n−1∏

j=1,2∤j,3∤j

(5 − 2 cos

2πj

12n

) 12n−1∏

j=1,2|j,3|j

(4 − 2 cos

2πj

12n

)

×12n−1∏

j=1,2|j,3|j

(8 − 2 cos

2πj

12n

) 12n−1∏

j=1,2|j,3|j

(7 − 2 cos

2πj

12n

)

×

6n−1∏j=1,2∤j

(8 − 2 cos 2πj

6n− 2 cos 2πj

3

) 6n−1∏j=1,2∤j

(4 − 2 cos 2πj

6n− 2 cos 2πj

3

)

6n−1∏j=1

(6 − 2 cos 2πj

6n− 2 cos πj

3

) .

Thus,

τ(C12n(n, 2n, 3n)) =1

12n

12n−1∏

j=1

(5 − 2 cos

2πj

12n

) 2n−1∏

j=1

(5 − 2 cos 2πj

2n

) (4 − 2 cos 2πj

2n

)(8 − 2 cos 2πj

2n

) (7 − 2 cos 2πj

2n

)

×6n−1∏

j=1

8 − 2 cos 2πj6n

5 − 2 cos 2πj6n

4n−1∏

j=1

7 − 2 cos 2πj4n

5 − 2 cos 2πj4n

×

6n−1∏j=1

(8 − 2 cos 2πj

6n− 2 cos 2πj

3

) 3n−1∏j=1

(4 − 2 cos 2πj

3n− 2 cos 4πj

3

)

6n−1∏j=1

(6 − 2 cos 2πj

6n− 2 cos πj

3

) 3n−1∏j=1

(8 − 2 cos 2πj

3n− 2 cos 4πj

3

) .


So, we get that

τ(C12n(n, 2n, 3n)) =1

12n

12n−1∏

j=1

(5 − 2 cos

2πj

12n

) 2n−1∏

j=1

(5 − 2 cos 2πj

2n

) (4 − 2 cos 2πj

2n

)(8 − 2 cos 2πj

2n

) (7 − 2 cos 2πj

2n

)

×6n−1∏

j=1

8 − 2 cos 2πj6n

5 − 2 cos 2πj6n

4n−1∏

j=1

7 − 2 cos 2πj4n

5 − 2 cos 2πj4n

×

6n−1∏j=1,3|j

(9 − 2 cos 2πj

6n

) 6n−1∏j=1,3|j

(6 − 2 cos 2πj

6n

)

6n−1∏j=1,3|j

(7 − 2 cos 2πj

6n

) 6n−1∏j=1,3|j

(4 − 2 cos 2πj

6n

)

×

3n−1∏j=1

(6 − 2 cos 2πj

3n− 4 cos2 2πj

3

)

3n−1∏j=1

(10 − 2 cos 2πj

3n− 4 cos2 2πj

3

) ,

i.e.,

τ(C12n(n, 2n, 3n)) =1

12n

12n−1∏

j=1

(5 − 2 cos

2πj

12n

) 2n−1∏

j=1

(5 − 2 cos 2πj

2n

) (4 − 2 cos 2πj

2n

)(8 − 2 cos 2πj

2n

) (7 − 2 cos 2πj

2n

)

×6n−1∏

j=1

8 − 2 cos 2πj6n

5 − 2 cos 2πj6n

4n−1∏

j=1

7 − 2 cos 2πj4n

5 − 2 cos 2πj4n

×

6n−1∏j=1

(9 − 2 cos 2πj

6n

) 6n−1∏j=1

(6−2 cos 2πj6n )

(9−2 cos 2πj6n )

6n−1∏j=1

(7 − 2 cos 2πj

6n

) 6n−1∏j=1

(4−2 cos 2πj6n )

(7−2 cos 2πj6n )

×

3n−1∏j=1,3|j

(5 − 2 cos 2πj

3n

) 3n−1∏j=1,3|j

(2 − 2 cos 2πj

3n

)

3n−1∏j=1,3|j

(9 − 2 cos 2πj

3n

) 3n−1∏j=1,3|j

(6 − 2 cos 2πj

3n

)

Furthermore,

τ(C12n(n, 2n, 3n)) =1

12n

12n−1∏

j=1

(5 − 2 cos

2πj

12n

) 2n−1∏

j=1

(5 − 2 cos 2πj

2n

) (4 − 2 cos 2πj

2n

)(8 − 2 cos 2πj

2n

) (7 − 2 cos 2πj

2n

)

×6n−1∏

j=1

8 − 2 cos 2πj6n

5 − 2 cos 2πj6n

4n−1∏

j=1

7 − 2 cos 2πj4n

5 − 2 cos 2πj4n


×

6n−1∏j=1

(9 − 2 cos 2πj

6n

) 2n−1∏j=1

(6−2 cos 2πj6n )

(9−2 cos 2πj6n )

6n−1∏j=1

(7 − 2 cos 2πj

6n

) 2n−1∏j=1

(4−2 cos 2πj6n )

(7−2 cos 2πj6n )

×

3n−1∏j=1

(5 − 2 cos 2πj

3n

) 3n−1∏j=1

(2−2 cos 2πj3n )

(5−2 cos 2πj3n )

3n−1∏j=1

(9 − 2 cos 2πj

3n

) 3n−1∏j=1

(6−2 cos 2πj3n )

(9−2 cos 2πj3n )

.

We get that

τ(C12n(n, 2n, 3n)) =1

12n

12n−1∏

j=1

(5 − 2 cos

2πj

12n

) 2n−1∏

j=1

(5 − 2 cos 2πj

2n

) (4 − 2 cos 2πj

2n

)(8 − 2 cos 2πj

2n

) (7 − 2 cos 2πj

2n

)

×6n−1∏

j=1

8 − 2 cos 2πj6n

5 − 2 cos 2πj6n

4n−1∏

j=1

7 − 2 cos 2πj4n

5 − 2 cos 2πj4n

×

6n−1∏j=1

(9 − 2 cos 2πj

6n

) 2n−1∏j=1

(6−2 cos 2πj6n )

(9−2 cos 2πj6n )

6n−1∏j=1

(7 − 2 cos 2πj

6n

) 2n−1∏j=1

(4−2 cos 2πj6n )

(7−2 cos 2πj6n )

×

3n−1∏j=1

(5 − 2 cos 2πj

3n

) n−1∏j=1

(2−2 cos 2πjn )

(5−2 cos 2πjn )

3n−1∏j=1

(9 − 2 cos 2πj

n

) n−1∏j=1

(6−2 cos 2πjn )

(9−2 cos 2πjn )

.

Thus,

τ(C12n(n, 2n, 3n)) =1

12n× U2

12n−1

(√7

4

)×U2

2n−1

(√74

)× U2

2n−1

(√32

)

U22n−1

(32

)× U2

2n−1

(√52

)

×U2

4n−1

(32

)× U2

6n−1

(√52

)

U26n−1

(√72

)× U2

6n−1

(√72

)

×U2

6n−1

(√114

)× U2

2n−1

(32

)× U2

2n−1

(√2)× U2

3n−1

(74

)× U2

n−1

(114

)n2

U26n−1

(32

)× U2

2n−1

(√32

)× U2

n−1

(√74

)× U2

3n−1

(√114

)× U2

n−1

(√2) ,

which implies that

τ(C12n(n, 2n, 3n)) =n

12

(√

7

4+

√3

4

)4n

+

(√7

4−√

3

4

)4n

− 1

2


× n

12

(√

7

4+

√3

4

)2n

+

(√7

4−√

3

4

)2n

+ 1

2

× n

12

[(√5

2+

√3

2

)2n

+(√5

2−√

3

2

)2n]2

× n

12

[(√2 + 1

)n

+(√

2 − 1)n]2

× n

12

(√

11

4+

√7

4

)2n

+

(√11

4−√

7

4

)2n

− 1

2

,

where (6), (8) , (9) and (10) are used to derive the last two steps. 2Theorem 3.4 For the spanning trees of C6n with three jumps 1, 3n, 6n, we have

τ(C12n(1, 3n, 6n)) =3n

4

(√

5

2+

√3

2

)6n

+

(√5

2−√

3

2

)6n

2

×3n

4

[(√2 + 1

)3n

+(√

2 − 1)3n]2


τ(C12n(1, 3n, 6n)) =1

12n

12n−1∏

j=1


)

=1

12n

12n−1∏

j=1

(6 − 2 cos

2πj

12n− 2 cos

6πjn

12n− 2 cos

12πjn

12n

)

=1

12n

12n−1∏

j=1

(6 − 2 cos

2πj

12n− 2 cos

πj

2− 2 cosπj

)

=1

12n

12n−1∏

j=1,2∤j

(8 − 2 cos

2πj

12n

) 12n−1∏

j=1,2|j

(4 − 2 cos

2πj

12n− 2 cos

πj

2

).

Whence,

τ (C12n(1, 3n, 6n)) =1

12n

12n−1∏

j=1

(8 − 2 cos

2πj

12n

) 6n−1∏

j=1

4 − 2 cos 2πj

12n− 2 cos πj

2

8 − 2 cos 2πj

6n

=1

12n

12n−1∏j=1

(8 − 2 cos 2πj

12n

)

6n−1∏j=1

(8 − 2 cos 2πj

6n

) ×6n−1∏

j=1,2∤j

(6 − 2 cos

2πj

6n

) 6n−1∏

j=1,2|j

(6 − 2 cos

2πj

6n

).


Thus,

τ(C12n(1, 3n, 6n)) =1

12n

12n−1∏j=1

(8 − 2 cos 2πj

12n

)

6n−1∏j=1

(8 − 2 cos 2πj

6n

)

6n−1∏

j=1

(6 − 2 cos

2πj

6n

)

×6n−1∏

j=1

2 − 2 cos 2πj6n

6 − 2 cos 2πj6n

=1

12n

12n−1∏j=1

(8 − 2 cos 2πj

12n

)

6n−1∏j=1

(8 − 2 cos 2πj

6n

)

6n−1∏

j=1

(6 − 2 cos

2πj

6n

)

×3n−1∏

j=1

2 − 2 cos 2πj3n

6 − 2 cos 2πj3n

,

which implies that,

τ(C12n(1, 3n, 6n)) =1

12n×U2

12n−1

(√52

)× (3n)2 × U2

6n−1(√

2)

U26n−1

(√52

)× U2

3n−1(√

2)

=3n

4×U2

12n−1

(√52

)× U2

6n−1(√

2)

U26n−1

(√52

)× U2

3n−1(√

2),

i.e.,

τ(C12n(1, 3n, 6n)) =3n

4

(√

5

2+

√3

2

)6n

+

(√5

2−√

3

2

)6n

2

×3n

4

[(√2 + 1

)3n

+(√

2 − 1)3n]2,

where (6), (8) , (9) and (10) are used to derive the last two steps. 2Theorem 3.5 For the spanning trees of C12n with three jumps 1, 3n, 4n , we have:

τ(C12n(1, 3n, 4n)) =n

12

(√

9

4+

√5

4

)4n

+

(√9

4−√

5

4

)4n

− 1

2

×

(√

7

4+

√3

4

)2n

+

(√7

4−√

3

4

)2n

+ 1

2


×

(√

3

2+

√1

2

)2n

+

(√3

2−√

1

2

)2n

2

×[(√

2 + 1)n

+(√

2 − 1)n]2

×

(√

11

4+

√7

4

)2n

+

(√11

4−√

7

4

)2n

− 1

2


τ(C12n(1, 3n, 4n)) =1

12n

12n−1∏

j=1


)

=1

12n

12n−1∏

j=1

(6 − 2 cos

2πj

12n− 2 cos

πj

2− 2 cos

2πj

3

)

=1

12n

12n−1∏

j=1,2∤j

(6 − 2 cos

2πj

12n− 2 cos

πj

3

)

×12n−1∏

j=1,2|j

(6 − 2 cos

2πj

12n− 2 cos

πj

3− 2 cos

πj

2

)

=1

12n

12n−1∏

j=1

(6 − 2 cos

2πj

12n− 2 cos

πj

3

)

×6n−1∏

j=1

6 − 2 cos 2πj6n

− 2 cosπj − 2 cos 4πj3

6 − 2 cos 2πj6n

− 2 cos 4πj3

=1

12n

12n−1∏

j=1,3∤j

(7 − 2 cos

2πj

12n

) 12n−1∏

j=1,3|j

(4 − 2 cos

2πj

12n

)

×6n−1∏

j=1

8 − 2 cos 2πj6n

− 2 cosπj − 2 cos2 2πj3

6n−1∏j=1

8 − 2 cos 2πj6n

− 2 cos2 2πj3

.

Thus,

τ (C12n(1, 3n, 4n)) =1

12n

12n−1∏

j=1

(7 − 2 cos

2πj

12n

) 4n−1∏

j=1

4 − 2 cos 2πj

4n

7 − 2 cos 2πj

4n

×

6n−1∏j=1,2∤j

(10 − 2 cos 2πj

6n− 4 cos2 2πj

3

) 6n−1∏j=1,2|j

(6 − 2 cos 2πj

6n− 4 cos2 2πj

3

)

6n−1∏j=1,3∤j

(7 − 2 cos 2πj

6n

) 6n−1∏j=1,3|j

(4 − 2 cos 2πj

6n

) .


We therefore get that

τ(C12n(1, 3n, 4n)) =1

12n

12n−1∏

j=1

(7 − 2 cos

2πj

12n

) 4n−1∏

j=1

4 − 2 cos 2πj4n

7 − 2 cos 2πj4n

×

6n−1∏j=1

(10 − 2 cos 2πj

6n− 4 cos2 2πj

3

) 3n−1∏j=1

6−2 cos 2πj3n

−4 cos2 2πj3

10−2 cos 2πj3n

−4 cos2 2πj3

6n−1∏j=1

(7 − 2 cos 2πj

6n

) 2n−1∏j=1

4−2 cos 2πj2n

7−2 cos 2πj2n

,

i.e.,

τ(C12n(1, 3n, 4n)) =1

12n

12n−1∏

j=1

(7 − 2 cos

2πj

12n

) 4n−1∏

j=1

4 − 2 cos 2πj4n

7 − 2 cos 2πj4n

×

6n−1∏j=1,3∤j

(9 − 2 cos 2πj

6n

) 6n−1∏j=1,3|j

(6 − 2 cos 2πj

6n

)

6n−1∏j=1

(7 − 2 cos 2πj

6n

) 2n−1∏j=1

4−2 cos 2πj2n

7−2 cos 2πj2n

×

6n−1∏j=1,3∤j

(5 − 2 cos 2πj

3n

) 6n−1∏j=1,3|j

(7 − 2 cos 2πj

3n

)

6n−1∏j=1,3∤j

(9 − 2 cos 2πj

3n

) 6n−1∏j=1,3|j

(6 − 2 cos 2πj

3n

) ,

which implies that

τ(C12n(1, 3n, 4n)) =1

12n

12n−1∏

j=1

(7 − 2 cos

2πj

12n

) 4n−1∏

j=1

4 − 2 cos 2πj4n

7 − 2 cos 2πj4n

×

6n−1∏j=1

(9 − 2 cos 2πj

6n

) 6n−1∏j=1

6−2 cos 2πj6n

9−2 cos 2πj6n

6n−1∏j=1

(7 − 2 cos 2πj

6n

) 2n−1∏j=1

4−2 cos 2πj2n

7−2 cos 2πj2n

×

3n−1∏j=1

(5 − 2 cos 2πj

3n

) 3n−1∏j=1

7−2 cos 2πj3n

5−2 cos 2πj3n

3n−1∏j=1

(9 − 2 cos 2πj

3n

) n−1∏j=1

6−2 cos 2πjn

9−2 cos 2πjn

.

Thus,

τ(C12n(1, 3n, 4n)) =1

12n×

U24n−1

(√32

)×U2

12n−1

(32

)

U24n−1

(√32

)

U22n−1

(√32

)×U2

6n−1

(32

)

U22n−1

(√32

)


×

U26n−1

(√114

)×U2

2n−1

(√

2

)

U22n−1

(√114

) ×n2×U2

3n−1

(√74

)

U23n−1

(√114

)

U2n−1

(√

2

)×U2

3n−1

(114

)

U2n−1

(√114

)

.

We have

τ(C12n(1, 3n, 4n)) =n

12

(√

9

4+

√5

4

)4n

+

(√9

4−√

5

4

)4n

− 1

2

×

(√

7

4+

√3

4

)2n

+

(√7

4−√

3

4

)2n

+ 1

2

×

(√

3

2+

√1

2

)2n

+

(√3

2−√

1

2

)2n

2

×[(√

2 + 1)n

+(√

2 − 1)n]2

×

(√

11

4+

√7

4

)2n

+

(√11

4−√

7

4

)2n

− 1

2

,

where (6), (8) , (9) and (10) are used to derive the last two steps. 2Theorem 3.6 For the spanning trees of C12n with three jumps 1, 2n, 3n, 6n , we have

τ (C12n(1, 2n, 3n, 6n)) =n

12

[(√11

4+

√7

4

)4n

+

(√11

4−√

7

4

)4n

− 1

]2

×[(√

11

4+

√7

4

)2n

+

(√11

4−√

7

4

)2n

+ 1

]2

×[(√

13

4+

√9

4

)4n

+

(√13

4−√

9

4

)4n

+ 1

]2

×[(√

7

2+

√5

2

)2n

+

(√7

2−√

5

2

)2n]2

×[(√

2 + 1)n

+(√

2 − 1)n]2

×[(√

7

4+

√3

4

)2n

+

(√7

4−√

3

4

)2n

+ 1

]2

.


Proof Let ε = e2πi12n . Applying Lemma 3.1, we get the required result. 2

Theorem 3.7 For the spanning trees of C12n with four jumps 1, 2n, 3n, 4n, we have

τ(C12n(1, 2n, 3n, 4n)) =n

12

(√

5

2+

√3

2

)6n

+

(√5

2−√

3

2

)6n

2

×

(√

5

2+

√3

2

)2n

+

(√5

2−√

3

2

)2n

+ 1

2

×

(√

7

2+

√5

2

)2n

+

(√7

2−√

5

2

)2n

− 1

2

×[(√

2 + 1)n

+(√

2 − 1)n]2

.


τ(C12n(1, 2n, 3n, 4n))

=1

12n

12n−1∏

j=1

(8 − ε−j − ε−2nj − ε−3nj − ε−4nj − εj − ε2nj − ε3nj − ε4nj

)

=1

12n

12n−1∏

j=1

(8 − 2 cos

2πjn

12n− 2 cos

6πjn

12n− 2 cos

8πjn

12n

)

=1

12n

12n−1∏

j=1

(8 − 2 cos

πj

3− 2 cos

πj

2− 2 cos

2πj

3

)

=1

12n

12n−1∏

j=1,2∤j

(8 − 2 cos

πj

3− 2 cos

πj

2− 2 cos

2πj

3

)

×12n−1∏

j=1,2|j

(8 − 2 cos

πj

3− 2 cos

πj

2− 2 cos

2πj

3

)

=1

12n

12n−1∏

j=1

(8 − 2 cos

πj

3− 2 cos

πj

2− 2 cos

2πj

3

)

×6n−1∏

j=1

8 − 2 cos πj3 − 2 cos πj

2 − 2 cos 2πj3

8 − 2 cos 2πj6n

− 2 cos 2πj3 − 2 cos 4πj

3

.

Thus,

τ(C12n(1, 2n, 3n, 4n)) =1

12n

12n−1∏

j=1,3∤j

(9 − 2 cos

2πj

12n− 2 cos

πj

3

)


×12n−1∏

j=1,3|j

(6 − 2 cos

2πj

12n− 2 cos

πj

3

)

×

6n−1∏j=1

(10 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3 − 2 cosπj

)

6n−1∏j=1

(10 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3

) ,

i.e.,

τ(C12n(1, 2n, 3n, 4n)) =1

12n

12n−1∏

j=

(9 − 2 cos

2πj

12n− 2 cos

πj

3

)

×4n−1∏

j=1

6 − 2 cos 2πj4n

− 2 cos πj3

9 − 2 cos 2πj4n

− 2 cos πj3

×

6n−1∏j=1

(10 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3 − 2 cosπj

)

6n−1∏j=1

(10 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3

) .

We get that

τ(C12n(1, 2n, 3n, 4n)) =1

12n

12n−1∏

j=1

(9 − 2 cos

2πj

12n− 2 cos

πj

3

)

×4n−1∏

j=1

6 − 2 cos 2πj4n

− 2 cos πj3

9 − 2 cos 2πj4n

− 2 cos πj3

×

6n−1∏j=1,2∤j

(12 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3

)

6n−1∏j=1,3∤j

(10 − 2 cos 2πj

6n

)

×

6n−1∏j=1,2|j

(8 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3

)

6n−1∏j=1,3|j

(4 − 2 cos 2πj

6n

) ,

which implies that

τ(C12n(1, 2n, 3n, 4n)) =1

12n

12n−1∏

j=1,2∤j,3∤j

(8 − 2 cos

2πj

12n

) 12n−1∏

j=1,2|j,3|j

(7 − 2 cos

2πj

12n

)


×12n−1∏

j=1,2∤j,3|j

(11 − 2 cos

2πj

12n

) 12n−1∏

j=1,2∤j,3|j

(10 − 2 cos

2πj

12n

)

×

4n−1∏j=1,2∤j

(8 − 2 cos 2πj

4n

) 4n−1∏j=1,2|j

(4 − 2 cos 2πj

4n

)

4n−1∏j=1,2∤j

(11 − 2 cos 2πj

4n

) 4n−1∏j=1,2|j

(7 − 2 cos 2πj

4n

)

×

6n−1∏j=1

(12 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3

)

6n−1∏j=1

(10 − 2 cos 2πj

6n

)

×

6n−1∏j=1

8−2 cos 2πj6n

−2 cos 2πj3 −4 cos2 2πj

3

12−2 cos 2πj6n


3

6n−1∏j=1

4−2 cos 2πj6n

10−2 cos 2πj6n

.

Thus,

τ(C12n(1, 2n, 3n, 4n)) =1

12n

12n−1∏

j=1

(8 − 2 cos

2πj

12n

)

×12n−1∏

j=1

(8 − 2 cos 2πj

12n

) (7 − 2 cos 2πj

12n

)(11 − 2 cos 2πj

12n

) (10 − 2 cos 2πj

12n

)

×12n−1∏

j=1

10 − 2 cos 2πj12n

]

8 − 2 cos 2πj12n

×12n−1∏

j=1

11 − 2 cos 2πj12n

8 − 2 cos 2πj12n

×

4n−1∏j=1

(8 − 2 cos 2πj

4n

) 4n−1∏j=1

4−2 cos 2πj4n

8−2 cos 2πj4n

4n−1∏j=1

(11 − 2 cos 2πj

4n

) 4n−1∏j=1

7−2 cos 2πj4n

11−2 cos 2πj4n

×

6n−1∏j=1

(12 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3

)

6n−1∏j=1

(10 − 2 cos 2πj

6n

)

×

6n−1∏j=1

8−2 cos 2πj6n


3

12−2 cos 2πj6n


3

6n−1∏j=1

4−2 cos 2πj6n

10−2 cos 2πj6n

.

We get that


τ(C12n(1, 2n, 3n, 4n)) =1

12n

12n−1∏

j=1

(8 − 2 cos

2πj

12n

)

×12n−1∏

j=1

(8 − 2 cos 2πj

12n

) (7 − 2 cos 2πj

12n

)(11 − 2 cos 2πj

12n

) (10 − 2 cos 2πj

12n

)

×12n−1∏

j=1

10 − 2 cos 2πj12n

8 − 2 cos 2πj12n

12n−1∏

j=1

11 − 2 cos 2πj12n

8 − 2 cos 2πj12n

×

4n−1∏j=1

(8 − 2 cos 2πj

4n

) 4n−1∏j=1

4−2 cos 2πj4n

8−2 cos 2πj4n

4n−1∏j=1

(11 − 2 cos 2πj

4n

) 4n−1∏j=1

7−2 cos 2πj4n

11−2 cos 2πj4n

×

6n−1∏j=1

(12 − 2 cos 2πj

6n− 2 cos 2πj

3 − 4 cos2 2πj3

)

6n−1∏j=1

(10 − 2 cos 2πj

6n

)

×

2n−1∏j=1

8−2 cos 2πj2n


3

12−2 cos 2πj2n


3

3n−1∏j=1

4−2 cos 2πj3n

10−2 cos 2πj3n

,

i.e.,

τ(C12n(1, 2n, 3n, 4n)) =1

12n

×U2

12n−1

(√114

)× U2

2n−1

(√114

)× U2

2n−1

(√72

)× U2

4n−1

(√72

)× U2

6n−1

(√134

)

U22n−1

(√72

)× U2

2n−1

(√134

)× U2

4n−1

(√114

)× U2

6n−1

(√114

)

×U2

6n−1

(√114

)× U2

2n−1

(√2)× U2

3n−1

(√74

)× U2

n−1

(√114

)× U2

2n−1

(√114

)

U22n−1

(√114

)× U2

n−1

(√74

)× U2

3n−1

(√114

)× U2

n−1

(√2)× U2

6n−1

(√114

)

So, we have

τ(C12n(1, 2n, 3n, 4n)) =n

12

(√

5

2+

√3

2

)6n

+

(√5

2−√

3

2

)6n

2

×

(√

5

2+

√3

2

)2n

+

(√5

2−√

3

2

)2n

+ 1

2


×

(√

7

2+

√5

2

)2n

+

(√7

2−√

5

2

)2n

− 1

2

×[(√

2 + 1)n

+(√

2 − 1)n]2

,

where (6), (8), (9) and (10) are used to derive the last two steps. 2Theorem 3.8 For the spanning trees of C6n with four jumps 1, 3n, 4n, 6n, we have

τ(C12n(1, 3n, 4n, 6n)) =n

12

(√

11

4+

√7

4

)8n

+

(√11

4−√

7

4

)8n

+ 1

2

×

(√

11

4+

√7

4

)2n

+

(√11

4−√

7

4

)2n

− 1

2

×

(√

3

2+

√5

2

)2n

+

(√3

2−√

5

2

)2n

2

×[(√

2 + 1)n

+(√

2 − 1)n]2

×

(√

7

4+

√3

4

)2n

+

(√7

4−√

3

4

)2n

+ 1

2

×

(√

13

4+

√9

4

)2n

+

(√13

4−√

9

4

)2n

+ 1

2

Proof Let ε = e2πi12n . Apply Lemma 3.1, we get the required result. 2

Theorem 3.9 For the spanning trees of C6n with four jumps 1, 2n, 3n, 4n, 6n, we have

τ(C12n(1, 2n, 3n, 4n, 6n)) =n

12×[(√

2 + 1)n

+(√

2 − 1)n]2

×[(√

2 +√

3)n

+(√

2 −√

3)n

− 1]2

×

(√

5

2+

√3

2

)2n

+

(√5

2−√

3

2

)2n

+ 1

2

×

(√

7

2+

√5

2

)2n

+

(√7

2−√

5

2

)2n

− 1

2

×

(√

7

2+

√5

2

)2n

+

(√7

2−√

5

2

)2n

2


×

(√

7

2+

√5

2

)4n

+

(√7

2−√

5

2

)4n

+ 1

2

Proof Let ε = e2πi12n . Applying Lemma 3.1, We get the required result. 2

§4. Conclusions

The number of spanning trees in graphs (networks) is an important invariant. The evaluation

of this number is not only interesting from a mathematical (computational) perspective, but

also, it is an important measure of reliability of a network and designing electrical circuits.

Some computationally hard problems such as the travelling salesman problem can be solved

approximately by using spanning trees. Due to the high dependence of the network design and

reliability on the graph theory we prove our results in Section 3.

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460.

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(1993).


Some Fixed Point Results for

Multivalued Nonexpansive Mappings in Banach Spaces

G.S.Saluja

(Department of Mathematics, Govt. Kaktiya P. G. College, Jagdalpur - 494001, Chhattisgarh, India)


Abstract: In this paper, we deal with the approximation of fixed point for multivalued

nonexpansive mappings via a new three-step algorithm which is independent and faster than

the iterative process discussed by Agarwal et al. [3] and establish some strong convergence

theorems and a weak convergence theorem in the setting of Banach spaces. Our results

extend and generalize the previous works from the existing literature.

Key Words: Multivalued nonexpansive mapping, three-step iteration scheme, fixed point,

weak and strong convergence, Banach space.

AMS(2010): 47H10, 54H25.

§1. Introduction

Throughout this paper, let X be a real Banach space with the norm ‖.‖. Let N denotes the set

of all positive integers and let F (T ) denotes the set of all fixed points of the mapping T .

Let K be a subset of X . A subset K is called proximal if for each x ∈ X , there exists

an element k ∈ K such that d(x, k) = inf{‖x − y‖ : y ∈ K} = d(x,K). It is well known that

a weakly compact convex subset of a Banach space and closed convex subsets of a uniformly

convex Banach space are Proximal.

We shall denote CB(K), C(K) and P (K) by the families of all nonempty closed and

bounded subsets, nonempty compact subsets and nonempty proximal subsets of K, respec-

tively. Let H denote the Hausdorff metric induced by the metric d of X , that is,

H(A,B) = max{supx∈A

d(x,B), supy∈B

d(y,A)}

for every A,B ∈ CB(X ), where d(x,B) = inf{‖x− y‖ : y ∈ B}.A multivalued mapping T : K → CB(K) is said to be a contraction if there exists a constant

b ∈ [0, 1) such that for any x, y ∈ K,

H(T x, T y) ≤ b ‖x− y‖,

1Received May 30, 2019, Accepted November 25, 2019.

Some Fixed Point Results for Multivalued Nonexpansive Mappings in Banach Spaces 49

and T is said to be nonexpansive if

H(T x, T y) ≤ ‖x− y‖,

for all x, y ∈ K. A point x ∈ K is called a fixed point of T if x ∈ T x.

Later, an interesting and rich fixed point theory for such maps was developed which has

applications in control theory, convex optimization, differential inclusion and economics (see

[5] and references cited therein). Moreover, the existence of fixed points for multivalued non-

expansive mappings in uniformly convex Banach spaces was proved by Lim [7]. Many authors

have studied the fixed point for multivalued mappings (e.g., see [4, 6, 8, 10, 15, 16, 19]).

In 2005, Sastry and Babu [11] obtained the convergence results from single valued mappings

to multivalued mappings by defining Ishikawa and Mann iterates for multivalued mappings with

a fixed point. They considered the following:

Let K be a nonempty convex subset of X , T : K → P (K) is a multivalued mapping with

p ∈ F (T ).

(i) The Mann iteration is defined by

{x1 = x ∈ K,

xn+1 = (1 − αn)xn + αnsn, n ∈ N,(1.1)

where {αn} is a real sequence in (0, 1) and sn ∈ T xn such that ‖sn − p‖ = d(p, T xn).

(ii) The Ishikawa iteration is defined by

x1 = x ∈ K,xn+1 = (1 − αn)xn + αnrn,

yn = (1 − βn)xn + βnsn, n ∈ N,

(1.2)

where {αn} and {βn} are real sequences in (0, 1), ‖sn − rn‖ = d(T xn, T yn) and ‖rn − p‖ =

d(T yn, T p) for sn ∈ T xn and rn ∈ T yn. They established some strong and weak convergence

results of the above iterates for multivalued nonexpansive mappings T under some appropriate

conditions.

In 2007, Panyanak [10] extended the results of Sastry and Babu [11] to a uniformly convex

Banach space and also modified the above Ishikawa iterative scheme as follows:

Let T : K → P (K) be a multi-valued mapping.

{x1 = x ∈ K,yn = (1 − βn)xn + βnzn, n ∈ N,

(1.3)

where {βn} is a real sequence in [0, 1], zn ∈ T xn and un ∈ T xn are such that ‖zn − un‖ =

50 G.S.Saluja

d(un, T xn) and ‖xn − un‖ = d(xn, F (T )), respectively, and

{x1 = x ∈ K,

xn+1 = (1 − αn)xn + αnz′n, n ∈ N,

(1.4)

where {αn} is a real sequence in [0, 1], z′n ∈ T xn and vn ∈ T xn are such that ‖z′n − vn‖ =

d(vn, T xn) and ‖yn − vn‖ = d(yn, F (T )), respectively and proved a convergence theorem of

Mann iterates for a mapping defined on a noncompact domain. Later in 2008, Song and

Wang [14] proved strong convergence theorems of Mann and Ishikawa iterates for multivalued

nonexpansive mappings under some appropriate control conditions. Furthermore, they also

gave an affirmative answer to Panyanak’s open question in [10].

Recently, Abbas and Nazir [1] introduced and studied the following iteration scheme: let

K be a nonempty subset of a Banach space X and T be a nonlinear mapping of K into itself.

Then the sequence {xn} in K is defined by

x1 = x ∈ K,xn+1 = (1 − αn)T yn + αnT zn,

yn = (1 − βn)T xn + βnT zn,

zn = (1 − γn)xn + γnT xn, n ∈ N,

(1.5)

where {αn}, {βn} and {γn} are real sequences in (0, 1). They showed that this process converges

faster than both Picard and the Agarwal et al. ([3]) and in support gave analytic proof by a

numerical example (for more details, see [1]).

Motivated by Sastry and Babu [11], Panyanak [10] and Song and Wang [14], we first give

a multivalued version of the iteration scheme (1.5) of Abbas and Nazir [1] and then study its

convergence analysis in the setting of Banach spaces. We define our iteration scheme as follows:

x1 = x ∈ K,xn+1 = (1 − αn)vn + αnwn,

yn = (1 − βn)un + βnwn,

zn = (1 − γn)xn + γnun, n ∈ N,

(1.6)

where {αn}, {βn} and {γn} are real sequences in (0, 1), un ∈ T xn, vn ∈ T yn and wn ∈ T zn

such that ‖wn − un‖ = d(T zn, T xn), ‖vn − wn‖ = d(T yn, T zn), ‖vn − un‖ = d(T yn, T xn),

‖un+1 − vn‖ = d(T xn+1, T yn) and ‖un+1 − wn‖ = d(T xn+1, T zn), respectively.

Now, we recall the following definitions.

Definition 1.1 A Banach space X is said to satisfy Opial condition [9] if for any sequence

{xn} in X, xn converges to x weakly it follows that

lim supn→∞

‖xn − x‖ < lim supn→∞

‖xn − y‖,

for all y ∈ X with y 6= x.


Examples of Banach spaces satisfying Opial condition are Hilbert spaces and all spaces

lp(1 < p <∞). On the other hand, Lp[0, 2π] with 1 < p 6= 2 fail to satisfy Opial condition.

Definition 1.2 A multivalued mapping T : K → P (X ) is called demiclosed at y ∈ K if for any

sequence {xn} in K weakly convergent to an element x and yn ∈ T xn strongly convergent to y,

we have y ∈ T x.

The following is the multivalued version of condition (I) of Senter and Dotson [13].

Definition 1.3 A multivalued nonexpansive mapping T : K → CB(K) where K a subset of Xis said to satisfy condition (I) if there exists a nondecreasing function f : [0,∞) → [0,∞) with

f(0) = 0, f(t) > 0 for all t ∈ (0,∞) such that d(x, T x) ≥ f(d(x, F (T ))) for all x ∈ K, where

F (T ) 6= ∅ is the fixed point set of the multivalued mapping T .

We need the following Lemmas to prove our main results.

Lemma 1.4 ([18]) Let {pn}, {qn}, {rn} be three sequences of nonnegative real numbers satis-

fying the following conditions:

pn+1 ≤ (1 + qn)pn + rn, n ≥ 0,∞∑

n=0

qn <∞,∞∑

n=0

rn <∞.

Then,

(1) limn→∞

pn exists;

(2) In addition, if lim infn→∞

pn = 0, then limn→∞

pn = 0.

Lemma 1.5 ([12]) Let E be a uniformly convex Banach space and 0 < α ≤ tn ≤ β < 1 for

all n ∈ N. Suppose further that {xn} and {yn} are sequences of E such that lim supn→∞

‖xn‖ ≤ a,

lim supn→∞

‖yn‖ ≤ a and limn→∞

‖tnxn+(1−tn)yn‖ = a hold for some a ≥ 0. Then limn→∞

‖xn−yn‖= 0.

Lemma 1.6 ([16]) Let T : K → P (K) be a multivalued mapping and PT (x) = {y ∈ T x :

‖x− y‖ = d(x, T x)}. Then the following are equivalent:

(1) x ∈ F (T );

(2) PT (x) = {x};(3) x ∈ F (PT ).

Moreover, F (T ) = F (PT ).

§2. Main Results

In this section we prove some strong and a weak convergence theorems using iteration scheme

(1.6). First, we need the following lemmas to prove main results.

Lemma 2.1 Let X be a real Banach space and K be a nonempty closed and convex subset of

52 G.S.Saluja

X . Let T : K → P (K) be a multivalued mapping such that F (T ) 6= ∅ and PT is a nonexpansive

mapping. Let {xn} be the sequence defined by (1.6), where {αn}, {βn} and {γn} are real

sequences in (0, 1). Then limn→∞

‖xn − p‖ exists for all p ∈ F (T ).

Proof Let p ∈ F (T ). Then p ∈ PT (p) = {p} by Lemma 1.6. It follows from (1.6) that

‖zn − p‖ ≤ (1 − γn)‖xn − p‖ + γn‖un − p‖≤ (1 − γn)‖xn − p‖ + γnH(PT (xn), PT (p))

≤ (1 − γn)‖xn − p‖ + γn‖xn − p‖= ‖xn − p‖. (2.1)

Again using (1.6) and (2.1), we obtain

‖yn − p‖ ≤ (1 − βn)‖un − p‖ + βn‖wn − p‖≤ (1 − βn)H(PT (xn), PT (p)) + βnH(PT (zn), PT (p))

≤ (1 − βn)‖xn − p‖ + βn‖zn − p‖≤ (1 − βn)‖xn − p‖ + βn‖xn − p‖= ‖xn − p‖. (2.2)

Now using (1.6), (2.1) and (2.2), we obtain

‖xn+1 − p‖ ≤ (1 − αn)‖vn − p‖ + αn‖wn − p‖≤ (1 − αn)H(PT (yn), PT (p)) + αnH(PT (zn), PT (p))

≤ (1 − αn)‖yn − p‖ + αn‖zn − p‖≤ (1 − αn)‖xn − p‖ + αn‖xn − p‖= ‖xn − p‖. (2.3)

It follows from Lemma 1.4 that limn→∞

‖xn − p‖ exists for each p ∈ F (T ). This completes the

proof. 2Lemma 2.2 Let X be a uniformly convex Banach space and K be a nonempty closed and

convex subset of X . Let T : K → P (K) be a multivalued mapping such that F (T ) 6= ∅ and

PT is a nonexpansive mapping. Let {xn} be the sequence defined by (1.6), where {αn}, {βn}and {γn} are real sequences in (0, 1). Then lim

n→∞d(xn, T xn) = 0, lim

n→∞d(xn, T yn) = 0 and

limn→∞

d(xn, T zn) = 0.

Proof From Lemma 2.1, limn→∞

‖xn − p‖ exists for each p ∈ F (T ). We suppose that

limn→∞

‖xn − p‖ = l for some l ≥ 0.

Since lim supn→∞

‖un − p‖ ≤ lim supn→∞

H(PT (xn), PT (p)) ≤ lim supn→∞

‖xn − p‖ = l, so

lim supn→∞

‖un − p‖ ≤ l. (2.4)


Again, since lim supn→∞

‖vn − p‖ ≤ lim supn→∞

H(PT (yn), PT (p)) ≤ lim supn→∞

‖yn − p‖ ≤ lim supn→∞

‖xn −p‖ = l, so

lim supn→∞

‖vn − p‖ ≤ l. (2.5)

Similarly,

lim supn→∞

‖wn − p‖ ≤ l. (2.6)

Applying Lemma 1.5, we get

limn→∞

‖un − vn‖ = 0 (2.7)

limn→∞

‖wn − un‖ = 0 (2.8)

and

limn→∞

‖vn − wn‖ = 0. (2.9)

Taking lim sup on both sides of (2.1) and (2.2), we obtain

lim supn→∞

‖zn − p‖ ≤ l (2.10)

and

lim supn→∞

‖yn − p‖ ≤ l. (2.11)

Also,

‖xn+1 − p‖ = ‖(1 − αn)vn + αnwn − p‖= ‖(vn − p) + αn(wn − vn)‖≤ ‖vn − p‖ + αn‖wn − vn‖≤ ‖vn − p‖ + ‖wn − vn‖

implies that

l ≤ lim infn→∞

‖vn − p‖. (2.12)

Combining (2.5) and (2.12), we obtain

limn→∞

‖vn − p‖ = l. (2.13)

Thus,

‖vn − p‖ ≤ ‖vn − wn‖ + ‖wn − p‖≤ ‖vn − wn‖ +H(PT (zn), PT (p))

≤ ‖vn − wn‖ + ‖zn − p‖

gives


‖zn − p‖ (2.14)

54 G.S.Saluja

and by virtue of (2.10), we obtain

limn→∞

‖zn − p‖ = l. (2.15)

Similarly,

‖wn − p‖ ≤ ‖wn − vn‖ + ‖vn − p‖≤ ‖wn − vn‖ +H(PT (yn), PT (p))

≤ ‖wn − vn‖ + ‖yn − p‖

gives


‖yn − p‖ (2.16)

and by virtue of (2.11), we obtain

limn→∞

‖yn − p‖ = l. (2.17)

Applying Lemma 1.5 once again, we obtain

limn→∞

‖xn − un‖ = 0. (2.18)

Notice that

‖xn − vn‖ ≤ ‖xn − un‖ + ‖un − vn‖.

Using (2.7) and (2.18), we obtain

limn→∞

‖xn − vn‖ = 0 (2.19)

and

‖xn − wn‖ ≤ ‖xn − un‖ + ‖un − wn‖.

Using (2.8) and (2.19), we obtain

limn→∞

‖xn − wn‖ = 0. (2.20)

Since d(xn, T xn) ≤ ‖xn − un‖, we have

limn→∞

d(xn, T xn) = 0. (2.21)

Again since d(xn, T yn) ≤ ‖xn − vn‖, we have

limn→∞

d(xn, T yn) = 0. (2.22)


Similarly, since d(xn, T zn) ≤ ‖xn − wn‖, we have

limn→∞

d(xn, T zn) = 0. (2.23)

This completes the proof. 2Now we shall prove some strong convergence theorems using iteration scheme (1.6) in real

Banach spaces.

Theorem 2.3 Let X be a real Banach space and K be a nonempty closed and convex subset

of X . Let T : K → P (K) be a multivalued mapping such that F (T ) 6= ∅ and PT is a non-

expansive mapping. Let {xn} be the sequence defined by (??), where {αn}, {βn} and {γn}are real sequences in (0, 1). Then {xn} converges strongly to a fixed point of T if and only if

lim infn→∞

d(xn, F (T )) = 0.

Proof The necessity is obvious. Conversely, suppose that lim infn→∞

d(xn, F (T )) = 0. As

proved in Lemma 1.6, we have

‖xn+1 − p‖ ≤ ‖xn − p‖

which gives

d(xn+1, F (T )) ≤ d(xn, F (T )).

This implies that limn→∞

d(xn, F (T )) exists by Lemma 1.4 and so by hypothesis, lim infn→∞

d(xn, F (T )) =

0. Therefore, we must have limn→∞

d(xn, F (T )) = 0.

Next, we have to show that {xn} is a Cauchy sequence in K. Let ε > 0 be arbitrary chosen.

Since limn→∞

d(xn, F (T )) = 0, there exists a constant n1 such that for all n ≥ n1 we have

d(xn, F (T )) <ε

4.

In particular, inf{‖xn1 − p‖ : p ∈ F (T )} < ε4 . There must exists a q ∈ F (T ) such that

‖xn1 − q‖ < ε

2.

Now for m,n ≥ n1, we have

‖xn+m − xn‖ ≤ ‖xn+m − q‖ + ‖xn − q‖≤ 2‖xn1 − q‖< 2

(ε2

)= ε.

Hence {xn} is a Cauchy sequence in a closed subset K of a Banach space X , and so it must

56 G.S.Saluja

converge in K. Let limn→∞

xn = q1. Now

d(q1, PT (q1)) ≤ ‖xn − q1‖ + d(xn, PT (xn)) +H(PT (xn), PT (q1))

≤ 2‖xn − q1‖ + ‖xn − un‖→ 0 as n→ ∞

which gives that d(q1, T q1) = 0. But PT is a nonexpansive mapping and so F (T ) is closed.

Therefore, q1 ∈ F (PT ) = F (T ). This shows that {xn} converges strongly to a point of F (T ).

This completes the proof. 2Theorem 2.4 Let X be a real Banach space and K be a nonempty compact convex subset of

X . Let T : K → P (K) be a multivalued mapping such that F (T ) 6= ∅ and PT is a nonexpansive

mapping. Let {xn} be the sequence defined by (??), where {αn}, {βn} and {γn} are real

sequences in (0, 1). Then {xn}, {yn} and {zn} converges strongly to a fixed point of T .

Proof By Lemma 2.2, we have limn→∞

d(xn, T xn) = 0. Since by hypothesis K be a nonempty

compact convex subset of X , so there exists a subsequence {xnk} of {xn} such that lim

k→∞‖xnk

−u‖ = 0 for some u ∈ K. Thus

d(u, PT (u)) ≤ ‖xnk− u‖ + d(xnk

, PT (xnk)) +H(PT (xnk

), PT (u))

≤ 2‖xnk− u‖ + ‖xnk

− unk‖

→ 0 as k → ∞.

This shows that u is a fixed of T . From Lemma 2.1, we get that limn→∞

‖xn − u‖ = 0. Again

from Lemma 2.2, we get that

‖yn − xn‖ = ‖(1 − βn)un + βnwn − xn‖≤ ‖un − xn‖ + βn‖wn − un‖≤ ‖un − xn‖ + ‖wn − un‖→ 0 as n→ ∞,

and

‖zn − xn‖ = ‖(1 − γn)xn + γnun − xn‖≤ γn‖un − xn‖≤ ‖un − xn‖→ 0 as n→ ∞.

It follows that limn→∞

‖yn − u‖ = 0 and limn→∞

‖zn − u‖ = 0. Thus the conclusion follows. This

completes the proof. 2Now, we apply Theorem 2.3 to obtain another strong convergence theorem in uniformly


convex Banach spaces satisfies condition (I) of Senter and Dotson [13].

Theorem 2.5 Let X be a uniformly convex Banach space and K be a nonempty closed and

convex subset of X . Let T : K → P (K) be a multivalued mapping satisfying condition (I) such

that F (T ) 6= ∅ and PT is a nonexpansive mapping. Let {xn} be the sequence defined by (1.6),

where {αn}, {βn} and {γn} are real sequences in (0, 1). Then {xn} converges strongly to a fixed

point of T .

Proof By Lemma 2.1, limn→∞

‖xn − p‖ exists for all p ∈ F (T ) and so the sequence {xn} is

bounded . Let limn→∞

‖xn − p‖ = r for some r > 0.

Now from Lemma 2.1, we know that

‖xn+1 − p‖ ≤ ‖xn − p‖

which implies that

infp∈F (T )

‖xn+1 − p‖ ≤ infp∈F (T )

‖xn − p‖,

and also d(xn+1, F (T )) ≤ d(xn, F (T )). And so, limn→∞

d(xn, F (T )) exists. By using condition

(I) and Lemma 2.2, we have

limn→∞

f(d(xn, F (T ))) ≤ limn→∞

d(xn, T xn) = 0.

That is,

limn→∞

f(d(xn, F (T ))) = 0.

Since f is a nondecreasing function and f(0) = 0, it follows that limn→∞

d(xn, F (T )) = 0. The

conclusion follows from Theorem 2.3. This completes the proof. 2Now, we prove the weak convergence theorem of the sequence {xn} defined by (1.6).

Theorem 2.6 Let X be a uniformly convex Banach space satisfying Opial’s condition and K be

a nonempty closed and convex subset of X . Let T : K → P (K) be a multivalued mapping such

that F (T ) 6= ∅ and PT is a nonexpansive mapping. Let {xn} be the sequence defined by (??),

where {αn}, {βn} and {γn} are real sequences in (0, 1). Let I − PT be demiclosed with respect

to zero. Then {xn} converges weakly to a fixed point of T .

Proof Let z ∈ F (T ). From Lemma 2.1, we know that limn→∞

‖xn − z‖ exists. Now we

prove that {xn} has a unique weak subsequential limit in F (T ). To prove this, let p1 and p2 be

weak limits of the subsequences {xni} and {xnj

} of {xn}, respectively. By (2.18), there exists

un ∈ T xn such that limn→∞

‖xn − un‖ = 0. Since I − PT is demiclosed with respect to zero,

therefore we obtain p1 ∈ F (T ). In the same way, we can prove that p2 ∈ F (T ).

Next, we prove uniqueness. For this, suppose that p1 6= p2. Then by Opial’s condition, we

58 G.S.Saluja

have

limn→∞

‖xn − p1‖ = limni→∞

‖xni− p1‖

< limni→∞

‖xni− p2‖

= limn→∞

‖xn − p2‖= lim

nj→∞‖xnj

− p2‖

< limnj→∞

‖xnj− p1‖

= limn→∞

‖xn − p1‖,

which is a contradiction. Hence {xn} converges weakly to a fixed point of T . This completes

the proof. 2Remark 2.7 Our results extend, generalize and improve several corresponding results from

the existing literature and iterative schemes discussed by Panyanak [10], Sastry and Babu [11],

Song and Wang [14] and Song and Cho [16] in the sense of faster iterative scheme.

Suzuki [17] introduced a condition on mappings called condition (C) which is weaker than

nonexpansiveness.

Recently, Abkar and Eslamian [2] introduced the definition of condition (C) for multi-

valued mapping. The definition is as follows.

Definition 2.8([2]) A multivalued mapping T : X → CB(X ) is said to satisfy condition (C)

provided that1

2d(x, T x) ≤ ‖x− y‖ ⇒ H(T x, T y) ≤ ‖x− y‖, x, y ∈ X .

The following result can be found in [2].

Lemma 2.9([2]) Let T : X → CB(X ) be a multi-valued mapping. If T is nonexpansive, then

T satisfies the condition (C).

We mention that there exist single-valued and multi-valued mappings satisfying the con-

dition (C) which are not nonexpansive.

Example 2.10([17]) Define a mapping T on [0.3] by

T (x) =

0, if x 6= 3,

1, if x = 3.

Then T is a single-valued mapping satisfying condition (C), but T is not nonexpansive.


Example 2.11([2]) Define a mapping T : [0, 5] → [0, 5] by

T (x) =

[0, x5 ], if x 6= 5,

{1}, if x = 5.

Then it is easy to show that T is a multi-valued mapping satisfying condition (C), but T is not

nonexpansive.

Now, we obtain some strong convergence results using iteration scheme (1.6) and condition

(C).

Theorem 2.12 Let X be a real Banach space and K be a nonempty closed and convex subset of

X . Let T : K → P (K) be a multivalued mapping such that F (T ) 6= ∅ and PT satisfies condition

(C). Let {xn} be the sequence defined by (1.6), where {αn}, {βn} and {γn} are real sequences in

(0, 1). Then {xn} converges strongly to a fixed point of T if and only if lim infn→∞

d(xn, F (T )) = 0.

Proof The proof of Theorem 2.12 immediately follows from Lemma 2.1 and Theorem 2.3.

This completes the proof. 2Theorem 2.13 Let X be a real Banach space and K be a nonempty closed and convex subset of

X . Let T : K → P (K) be a multivalued mapping such that F (T ) 6= ∅ and PT satisfies condition

(C). Let {xn} be the sequence defined by (1.6), where {αn}, {βn} and {γn} are real sequences

in (0, 1). If the following condition is satisfied:

(c1) there exists an increasing function ψ : [0,∞) → [0,∞) with ψ(r) > 0, ∀ r > 0 such

that

d(xn, T xn) ≥ ψ(d(xn, F (T )), (2.24)

then {xn} converges strongly to a fixed point of T .

Proof As in the proof of Lemma 2.2, we know that limn→∞

d(xn, T xn) = 0. Hence from

(2.21) we obtain limn→∞

d(xn, F (T )) = 0. The conclusion of Theorem 2.13 can be obtained from

Theorem 2.12 immediately. This completes the proof. 2§3. Concluding Remarks

In this paper, we study a new three-step iteration scheme for multivalued nonexpansive map-

pings in Banach spaces and establish some strong convergence theorems and a weak convergence

theorem under some appropriate conditions applying on the space. Our results extend and gen-

eralize several results from the current existing literature.

References

[1] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization

and feasibility problems, Mat. Vesnik, 66(2) (2014), 223-234.

[2] A. Abkar and M. Eslamian, Fixed point theorems for Suzuki generalized nonexpansive

60 G.S.Saluja

multivalued mappings in Banach spaces, Fixed Point Theory Appl., (2010) Article ID

457935, 10 pages.

[3] R. P. Agarwal, D. O’Regan and D. R. Sahu, Iterative construction of fixed points of nearly

asymptotically nonexpansive mappings, Nonlinear Convex Anal., 8(1) (2007), 61-79.

[4] N. A. Assad and W. A. Kirk, Fixed point theorems for set-valued mappings ocontractive

type, Pacific J. Math., 43 (1972), 553-562.

[5] L. Gorniewicz, Topologfixed Point Theory of Multivalued Mappings, Kluwer Academic Pub.

Dordrecht, Netherlands, (1999).

[6] W. A. Kirk, Transfinite methods in metric fixed point theory, Abstract and Applied Anal-

ysis, 5 (2003), 311-324.

[7] T. C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly

convex Banach space, Bull. Amer. Math. Soc., 80 (1974), 1123-1126.

[8] T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976),

179-182.

[9] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive

mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.

[10] B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach

spaces, Comp. Math. Appl., 54 (2007), 872-877.

[11] K. P. R. Sastry, and G. V. R. Babu, Convergence of Ishikawa iterates for a multivalued

mapping with a fixed point, Czechoslovak Math. J., 55 (2005), 817-826.

[12] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings,

J. Math. Anal. Appl., 158 (1991), 407-413.

[13] H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings,

Proc. Amer. Math. Soc., 44 (1974), 375-380.

[14] Y. S. Song and H. J. Wang, Erratum Mann and Ishikawa iterative processes for multivalued

mappings in Banach spaces, Comput. Math. Appl., 55 (2008), 2999-3002.

[15] Y. S. Song and H. J. Wang, Convergence of iterative algorithms for multivalued mappings

in Banach spaces, Nonlinear Anal., 70(4) (2009), 1547-1556.

[16] Y. S. Song and Y. J. Cho, Some notes on Ishikawa iteration for multivaled mappings, Bull.

Korean Math. Soc., 48(3) (2011), 575-584.

[17] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonex-

pansive mappings, J. Math. Anal. Appl., 340(2) (2008), 1088-1095.

[18] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa

iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.

[19] H. K. Xu, Multivalued nonexpansive mappings in Banach spaces, Nonlinear Anal., 43

(2001), 693-706.


Algebraic Properties of the Path Complexes of Cycles

Seyed Mohammad Ajdani

(Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran)


Abstract: Let G be a simple graph and ∆t (G) be a simplicial complex whose facets

correspond to the paths of length t (t ≥ 2) in G. It is shown that ∆t (Cn) is matroid, vertex

decomposable, shellable and Cohen-Macaulay if and only if n = t or n = t + 1, where Cn is

an n-cycle. As a consequence we show that if n = t or t + 1 then ∆t (Cn) is partitionable

and Stanley’s conjecture holds for K[∆t (Cn)].

Key Words: Vertex decomposable, simplicial complex, matroid, path.

AMS(2010): 13F20, 05E40, 13F55.

§1. Introduction

Let R = K[x1, · · · , xn], where K is a field. Fix an integer n ≥ t ≥ 2 and let G be a directed

graph. A sequence xi1 , · · · , xitof distinct vertices is called a path of length t if there are t− 1

distinct directed edges e1, · · · , et−1 where ej is a directed edge from xijto xij+1 . Then the path

ideal of G of length t is the monomial ideal It(G) = (xi1 · · ·xit: xi1 , · · · , xit

is a path of length

t in G in the polynomial ring R = K[x1, · · · , xn]. The distance d(x, y) of two vertices x and y

of a graph G is the length of the shortest path from x to y. The path complex ∆t(G) is defined

by

∆t(G) = 〈{xi1 , · · · , xit} : xi1 , · · · , xit

is a path of length t in G 〉.

Path ideals of graphs were first introduced by Conca and De Negri [3] in the context of

monomial ideals of linear type. Recently the path ideal of cycles has been extensively studied

by several mathematicians. In [9], it is shown that I2(Cn) is sequentially Cohen-Macaulay, if

and only if, n = 3 or n = 5. Generalizing this result, in [13], it is proved that It(Cn), (t > 2),

is sequentially Cohen-Macaulay, if and only if n = t or n = t+ 1 or n = 2t+ 1. Also, the Betti

numbers of the ideal It(Cn) and It(Ln) is computed explicitly in [1]. In particular, it has been

shown that

Theorem 1.1(Corollary 5.15, [1]) Let n, t, p and d be integers such that n ≥ t ≥ 2, n =

(t+ 1)p+ d, where p ≥ 0 and 0 ≤ d < (t+ 1). Then,

1Received June 17, 2019, Accepted November 26, 2019.

62 Seyed Mohammad Ajdani

(i) The projective dimension of the path ideal of a graph cycle Cn or line Ln is given by

pd (It(Cn)) =

2p, d 6= 0

2p− 1, d = 0pd (It(Ln)) =

2p− 1, d 6= t,

2p, d = t.

(ii) The regularity of the path ideal of a graph cycle Cn or line Ln is given by

reg (It(Cn)) = (t− 1)p+ d+ 1

reg (It(Ln)) =

p(t− 1) + 1, d < t,

p(t− 1) + t, d = t.

In [8] it has been shown that, ∆t(G) is a simplicial tree if G is a rooted tree and t ≥ 2.

One of interesting problems in combinatorial commutative algebra is the Stanley’s conjectures.

The Stanley’s conjectures are studied by many researchers. Let R be a Nn-graded ring and M

a Zn- graded R- module. Then, Stanley [10] conjectured that

depth (M) ≤ sdepth (M)

He also conjectured in [11] that each Cohen-Macaulay simplicial complex is partitionable. Her-

zog, Soleyman Jahan and Yassemi in [7] showed that the conjecture about partitionability is

a special case of the Stanley’s first conjecture. In this work, we study algebraic properties of

∆t(Cn). In Section 1, we recall some definitions and results which will be needed later. In

Section 3, we show that the following conditions are equivalent for all t > 2:

(i) ∆t(Cn) is matroid;

(ii) ∆t(Cn) is vertex decomposable;

(iii) ∆t(C n) is shellable;

(iv) ∆t(Cn) is Cohen-Macaulay;

(v) n = t or t+ 1.

(See Theorem 3.6).

In Section 4 as an application of our results we show that if n = t or t+ 1 then ∆t (Cn) is

partitionable and Stanley’s conjecture holds for K[∆t (Cn)].

§2. Preliminaries

In this section we recall some definitions and results which will be needed later.

Definition 2.1 A simplicial complex ∆ over a set of vertices V = {x1, · · · , xn}, is a collection

of subsets of V , with the property that:

(a) {xi} ∈ ∆ for all i;

(b) If F ∈ ∆, then all subsets of F are also in ∆ (including the empty set).

Algebraic Properties of the Path Complexes of Cycles 63

An element of ∆ is called a face of ∆ and complement of a face F is V \F and it is denoted

by F c. Also, the complement of the simplicial complex ∆ = 〈F1, · · · , Fr〉 is ∆c = 〈F c1 , · · · , F c

r 〉.The dimension of a face F of ∆, dimF , is |F | − 1 where, |F | is the number of elements of F

and dim ∅ = −1. The faces of dimensions 0 and 1 are called vertices and edges, respectively.

A non-face of ∆ is a subset F of V with F /∈ ∆. we denote by N (∆), the set of all minimal

non-faces of ∆. The maximal faces of ∆ under inclusion are called facets of ∆. The dimension

of the simplicial complex ∆, dim∆, is the maximum of dimensions of its facets. If all facets of

∆ have the same dimension, then ∆ is called pure.

Let F(∆) = {F1, · · · , Fq} be the facet set of ∆. It is clear that F(∆) determines ∆

completely and we write ∆ = 〈F1, · · · , Fq〉. A simplicial complex with only one facet is called

a simplex. A simplicial complex Γ is called a subcomplex of ∆, if F(Γ) ⊂ F(∆).

For v ∈ V , the subcomplex of ∆ obtained by removing all faces F ∈ ∆ with v ∈ F is

denoted by ∆ \ v. That is,

∆ \ v = 〈F ∈ ∆ : v /∈ F 〉.

The link of a face F ∈ ∆, denoted by link∆(F ), is a simplicial complex on V with the

faces, G ∈ ∆ such that, G ∩ F = ∅ and G ∪ F ∈ ∆. The link of a vertex v ∈ V is simply

denoted by link∆(v).

link∆(v) ={F ∈ ∆ : v /∈ F, F ∪ {v} ∈ ∆

}.

Let ∆ be a simplicial complex over n vertices {x1, · · · , xn}. For F ⊂ {x1, · · · , xn}, we set:

xF =∏

xi∈F

xi.

We define the facet ideal of ∆, denoted by I(∆), to be the ideal of S generated by {xF : F ∈F(∆)}. The non-face ideal or the Stanley-Reisner ideal of ∆, denoted by I∆, is the ideal of

S generated by square-free monomials {xF : F ∈ N (∆)}. Also we call K[∆] := S/I∆ the

Stanley-Reisner ring of ∆.

Definition 2.2 A simplicial complex ∆ on {x1, · · · , xn} is said to be a matroid if, for any two

facets F and G of ∆ and any xi ∈ F , there exists a xj ∈ G such that (F \ {xi}) ∪ {xj} is a

facet of ∆.

Definition 2.3 A simplicial complex ∆ is recursively defined to be vertex decomposable, if it is

either a simplex, or else has some vertex v so that

(a) Both ∆ \ v and link∆(v) are vertex decomposable, and

(b) No face of link∆(v) is a facet of ∆ \ v.A vertex v which satisfies in condition (b) is called a shedding vertex.

Definition 2.4 A simplicial complex ∆ is shellable, if the facets of ∆ can be ordered F1, · · · , Fs

such that, for all 1 ≤ i < j ≤ s, there exists some v ∈ Fj \ Fi and some l ∈ {1, · · · , j − 1} with

Fj \ Fl = {v}.


A simplicial complex ∆ is called disconnected, if the vertex set V of ∆ is a disjoint union

V = V1 ∪ V2 such that no face of ∆ has vertices in both V1 and V2. Otherwise ∆ is connected.

It is well-known that

matroid =⇒ vertex decomposable =⇒ shellable =⇒ Cohen-Macaulay

Definition 2.5 For a given simplicial complex ∆ on V , we define ∆∨, the Alexander dual of

∆, by

∆∨ = {V \ F : F /∈ ∆}.

It is known that for the complex ∆ one has I∆∨ = I(∆c). Let I 6= 0 be a homogeneous

ideal of S and N be the set of non-negative integers. For every i ∈ N ∪ {0}, one defines:

tSi (I) = max{j : βSi,j(I) 6= 0}

where βSi,j(I) is the i, j-th graded Betti number of I as an S-module. The Castelnuovo-Mumford

regularity of I is given by:

reg(I) = sup{tSi (I) − i : i ∈ Z}.

We say that the ideal I has a d-linear resolution, if I is generated by homogeneous polynomials

of degree d and βSi,j(I) = 0, for all j 6= i + d and i ≥ 0. For an ideal which has a d-linear

resolution, the Castelnuovo-Mumford regularity would be d. If I is a graded ideal of S, we

write (Id) for the ideal generated by all homogeneous polynomials of degree d belonging to I.

Definition 2.6 A graded ideal I is componentwise linear if (Id) has a linear resolution for all

d.

Also, we write I[d] for the ideal generated by the squarefree monomials of degree d belonging

to I.

Definition 2.7 A graded S-module M is called sequentially Cohen-Macaulay (over K), if there

exists a finite filtration of graded S-modules,

0 = M0 ⊂M1 ⊂ · · · ⊂Mr = M

such that each Mi/Mi−1 is Cohen-Macaulay, and the Krull dimensions of the quotients are

increasing:

dim(M1/M0) < dim(M2/M1) < · · · < dim(Mr/Mr−1).

The Alexander dual, allows us to make a bridge between (sequentially) Cohen-Macaulay

ideals and (componetwise) linear ideals.

Definition 2.8(Alexander Duality) For a square-free monomial ideal I = (M1, · · · ,Mq) ⊂ S =


K[x1, · · · , xn], the Alexander dual of I, denoted by I∨, is defined to be

I∨ = PM1 ∩ · · · ∩ PMq

where, PMiis prime ideal generated by {xj : xj |Mi}.

Theorem 2.9(Proposition 8.2.20, [6]; Theorem 3, [4]) Let I be a square-free monomial ideal

in S = K[x1, · · · , xn].

(i) The ideal I is componentwise linear ideal if and only if S/I∨ is sequentially Cohen-

Macaulay;

(ii) The ideal I has a q-linear resolution if and only if S/I∨ is Cohen-Macaulay of dimen-

sion n− q.

Remark 2.10 Two special cases, we will be considering in this paper, are when G is a cycle

Cn, or a line graph Ln on vertices {x1, · · · , xn} with edges

E (Cn) ={{x1, x2}, {x2, x3}, · · · , {xn−1, xn}, {xn, x1}

};

E (Ln) ={{x1, x2}, {x2, x3}, · · · , {xn−1, xn}

}.

§3. Vertex Decomposability Path Complexes of Cycles

As the main result of this section, it is shown that ∆t (Cn) is matroid, vertex decomposable,

shellable and Cohen-Macaualay if and only if n = t or n = t + 1. For the proof we shall need

the following lemmas and propositions.

Lemma 3.1 Let ∆t(Ln) be a simplicial complex on {x1, · · · , xn} and 2 ≤ t ≤ n. Then ∆t(Ln)

is vertex decomposable.

Proof If t = n, then ∆n(Ln) is a simplex which is vertex decomposable. Let 2 ≤ t < n

then one has

∆t(Ln) = 〈{x1, · · · , xt}, {x2, · · · , xt+1}, · · · , {xn−t+1, · · · , xn}〉.

So ∆t(Ln) \ xn = 〈{x1, · · · , xt}, {x2, · · · , xt+1}, · · · , {xn−t, · · · , xn−1}〉. Now we use induction

on the number of vertices of Ln and by induction hypothesis ∆t(Ln)\xn is vertex decomposable.

On the other hand, it is clear that link∆t(Ln){xn} = 〈{xn−t+1, · · · , xn−1}〉. Thus link∆t(Ln){xn}is a simplex which is not a facet of ∆t(Ln) \ xn. Therefore ∆t(Ln) is vertex decomposable. 2Lemma 3.2 Let ∆2(Cn) be a simplicial complex on {x1, · · · , xn}. Then ∆2(Cn) is vertex

decomposable.

Proof Since ∆2(Cn) = 〈{x1, x2}, {x2, x3}, · · · , {xn−1, xn}, {xn, x1}〉, we have

∆2(Cn) \ xn = 〈{x1, , x2}, {x2, x3}, · · · , {xn−2, xn−1}〉.


By lemma 3.1 ∆2(Cn)\xn is vertex decomposable. Also it is trivial that link∆2(Cn){xn} =

〈{xn−1}, {x1}〉 is vertex decomposable and no face of link∆2(Cn){xn} is a facet of ∆2(Cn) \ xn.

Therefore ∆2(Cn) is vertex decomposable. 2Lemma 3.3 Let ∆t(Cn) be a simplicial complex on {x1, · · · , xn} and 3 ≤ t ≤ n − 2. Then

∆t(Cn) is not Cohen-Macaulay.

Proof It suffices to show that I∆t(Cn)∨ has not a linear resolution. Since I∆t(Cn)∨ =

I(∆t(Cn)c) then one can easily check that I∆t(Cn)∨ = In−t(Cn). By Theorem 1.1 we have

reg(I∆t(Cn)∨) = (n− t− 1)p+ d+ 1.

Since 3 ≤ t ≤ n− 2 then one has reg(I∆t(Cn)∨) 6= n− t and by Theorem 2.9 ∆t(Cn) is not

Cohen-Macaulay. 2Proposition 3.4 Let ∆t(Cn) be a simplicial complex on {x1, · · · , xn} and t ≥ 3. Then ∆t(Cn)

is vertex decomposable if and only if n = t or t+ 1.

Proof By Lemma 3.3 it suffices to show that if n = t or t + 1, then ∆t(Cn) is vertex

decomposable. If n = t, then ∆n(Cn) is a simplex which is vertex decomposable. If t = n− 1,

then we have

∆n−1(Cn) = 〈{x1, · · · , xn−1}, {x2, · · · , xn}, {x3, · · · , xn, x1}, · · · , {xn, x1, · · · , xn−2}〉.

Now we use induction on the number of vertices of Cn and show that ∆n−1(Cn) is vertex

decomposable. It is clear that ∆n−1(Cn) \ xn = 〈{x1, · · · , xn−1}〉 is a simplex which is vertex

decomposable.

On the other hand,

link∆n−1(Cn){xn} = 〈{x1, · · · , xn−2}, · · · , {xn−1, x1, · · · , xn−3}〉 = ∆n−2(Cn−1).

By induction hypothesis link∆n−1(Cn){xn} is vertex decomposable. It is easy to see that

no face of link∆n−1(Cn){xn} is a facet of ∆n−1(Cn) \ xn. Therefore ∆n−1(Cn) is vertex decom-

posable. 2Proposition 3.5 ∆2(Cn) is a matroid if and only if n = 3 or 4.

Proof If n = 3 or 4, then it is easy to see that ∆2(Cn) is a matroid. Now we prove the

converse. It suffices to show that ∆2(Cn) is not a matroid for all n ≥ 5. We consider two

facets {x1, x2} and {xn−1, xn}. Then we have ({x1, x2} \ {x1}) ∪ {xn−1} = {x2, xn−1} and

({x1, x2}\{x1})∪{xn} = {x2, xn}. Since {x2, xn−1} and {x2, xn} are not the facets of ∆2(Cn).

So ∆2(Cn) is not matroid for all n ≥ 5. 2For the simplicial complexes one has the following implication:

Matroid ⇒ vertex decomposable ⇒ shellable ⇒ Cohen-Macaulay


Note that these implications are strict, but by the following theorem, for path complexes,

the reverse implications are also valid.

Theorem 3.6 Let t ≥ 3. Then the following conditions are equivalent:

(i) ∆t(Cn) is matroid;

(ii) ∆t(Cn) is vertex decomposable;

(iii) ∆t(Cn) is shellable;

(iv) ∆t(Cn) is Cohen-Macaulay;

(v) n = t or t+ 1.

Proof (i) =⇒ (ii), (ii) =⇒ (iii) and (iii) =⇒ (iv) is well-known.

(iv) =⇒ (v) follows from Lemma 3.3 and Proposition 3.4.

(v) =⇒ (i): If n = t, then ∆t(Cn) is a simplex which is a matroid. If n = t+ 1, then

∆t(Cn) = 〈{x1, · · · , xt}, {x2, · · · , xt+1}, {x3, · · · , xt+1, x1}, · · · , {xt+1, x1, · · · , xt−1}〉.

For any two facets F and G of ∆t(Cn) one has | F ∩G |= t− 1. We claim that for any two

facets F and G of ∆t(Cn) and any xi ∈ F , there exists a xj ∈ G such that (F \{xi})∪{xj} is a

facet of ∆t(Cn). We have to consider two cases. If xi ∈ F and xi /∈ G, then we choose xj ∈ G

such that xj /∈ F . Thus (F \ {xi}) ∪ {xj} = G which is a facet of ∆t(Cn).

For other case, if xi ∈ F and xi ∈ G, then we choose xj ∈ G such that xj is the same xi.

Therefore (F \ {xi}) ∪ {xi} = F is a facet of ∆t(Cn) which completes the proof. 2§4. Stanley Decompositions

Let R be any standard graded K- algebra over an infinite field K, i.e, R is a finitely gener-

ated graded algebra R =⊕

i≥0 Ri such that R0 = K and R is generated by R1. There are

several characterizations of the depth of such an algebra. We use the one that depth (R) is

the maximal length of a regular R- sequence consisting of linear forms. Let xF = ⊓i∈Fxi be

a squarefree monomial for some F ⊆ [n] and Z ⊆ {x1, · · · , xn}. The K- subspace xFK[Z]

of S = K[x1, · · · , xn] is the subspace generated by monomials xFu, where u is a monomial in

the polynomial ring K[Z]. It is called a square free Stanley space if {xi : i ∈ F} ⊆ Z. The

dimension of this Stanley space is |Z|. Let ∆ be a simplicial complex on {x1, · · · , xn}. A square

free Stanley decomposition D of K[∆] is a finite direct sum⊕

i uiK[Zi] of squarefree Stanley

spaces which is isomorphic as a Zn- graded K- vector space to K[∆], i.e.

K[∆] ∼=⊕

i

uiK[Zi].

We denote by sdepth (D) the minimal dimension of a Stanley space in D and define sdepth (K[∆])

= max{sdepth (D)}, where D is a Stanley decomposition of K[∆]. Stanley conjectured in [10]


the upper bound for the depth of K[∆] holding with

depth (K[∆]) ≤ sdepth (K[∆]).

Also we recall another conjecture of Stanley. Let ∆ be again a simplicial complex on

{x1, · · · , xn} with facets G1, · · · , Gt. The complex ∆ is called partitionable if there exists a

partition ∆ =⋃t

i=1[Fi, Gi] where Fi ⊆ Gi are suitable faces of ∆. Here the interval [Fi, Gi] is

the set of faces {H ∈ ∆ : Fi ⊆ H ⊆ Gi}. In [11] and [12] respectively Stanley conjectured each

Cohen-Macaulay simplicial complex is partitionable. This conjecture is a special case of the

previous conjecture. Indeed, Herzog, Soleyman Jahan and Yassemi [7] proved that for Cohen-

Macaulay simplicial complex ∆ on {x1, . . . , xn} we have that depth (K[∆]) ≤ sdepth (K[∆]) if

and only if ∆ is partitionable. Since each vertex decomposable simplicial complex is shellable

and each shellable complex is partitionable. Then as a consequence of our results, we obtain

Corollary 3.1 If n = t or t + 1 then ∆t (Cn) is partitionable and Stanley’s conjecture holds

for K[∆t (Cn)].

Acknowledgment. The author is deeply grateful to the referee for careful reading of the

manuscript and helpful suggestions.

References

[1] A. Aliloee and S. Faridi, Betti numbers of the path ideals of cycles and lines, preprint

(2011).

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Amsterdam, (1995).

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Algebra, 211 (1999), no. 2, 599-624.

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Pure and Applied Algebra, 130 (1998), 265-275.

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[6] J. Herzog and T. Hibi, Componentwise linear ideals, Nagoya Math. J., 153 (1999), 141-153.

[7] J. Herzog, A. Soleyman Jahan and S. Yassemi, Stanley decompositions and partitionable

simplicial complexes, J. Alger. Comb., 27(2008), 113-125

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38 (2010), 1725-1742.

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Math. Soc., (2007). No.8, 2327-2337.

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No 2, 175-193.

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ematics, 41, Birkhauser Boston 1996.


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matics: Frontiers and Perspectives, 295-319, Amer. Math. Soc., Providence, RI 2000.

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sigraph, Southeast Asian Bull. Matt., 34 (5)(2010), 953-958.

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Math. Soc., 137 (2009), 3235-3246.


Some Classes of Analytic Functions with q-Calculus

Hamid Shamsan1, Fuad S. M. Alsarari2 and S. Latha1

1. Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysuru-570 005, India

2. Department of Mathematics and Statistics, College of Sciences, Yanbu, Taibah University, Saudi Arabia


Abstract: We study the estimates for the Second q-Hankel determinant of analytic func-

tions in a class which unifies a number of classes studied previously by Darus, Ramreddy,

Ravichandran, Yang and others. Our class includes q-convex and q-starlike functions. Also

we study the estimate for q-Toeplitz determinants whose elements are the coefficients an for

f in close-to-q-convex functions.

Key Words: Second Hankel determinant, subordination, q-starlike and q-convex functions,

close-to-q-convex, Toeplitz matrices.

AMS(2010): 30C45. 30C50.

§1. Introduction

The Hankel determinants Hq(n) of Taylor’s coefficients of function f ∈ A where A denotes the

class of functions of the form

f(z) = z +

∞∑

n=2

anzn, (1.1)

which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1}. is defined by

Hq(n) :=

∣∣∣∣∣∣∣∣∣∣∣∣

an an+1 · · · an+q−1

an+1 an+2 · · · an+q

...... · · ·

...

an+q−1 an+q · · · an+2q−2

∣∣∣∣∣∣∣∣∣∣∣∣

where (a1 = 1, n = q ∈ N). Hankel matrices (and determinants) play an important role in

several branches of mathematics and have many applications [11]. H2(1) is the classical Fekete-

Szego functional. Fekete-Szego in [4] found the maximum value of H2(1). Pommerenke in [16]

proved that the Hankel determinant of univalent functions satisfy

|Hq(n)| < Kn−( 12 +β)q+ 3

2 (n = 1, 2, · · · , q = 2, 3, · · · ),

1Received July 31, 2019, Accepted December 1, 2019.

Some Classes of Analytic Functions with q-Calculus 71

where β > 14000 and K depends only on q.

Hayman [8] showed that

|H2(n)| = |anan+2 − a2n+1| < An

12 , n = 2, 3, · · · ,

where A is an absolute constant for a really mean univalent functions. Hankel determinants

are useful in showing that a function of bounded characteristic in U , i.e, a function which

is ratio of tow bounded analytic functions with its Laurent series around the origin having

integral coefficients, is rational. In recent years, several authors investigated bounds for the

Hankel determinant belonging tow various subclasses of univalent and multivalent functions

in a class which unifies a number of classes studied earlier by Deepak Bansal, K. I. Noor, T.

Yavuz, Sarika Verma, Shigeyoshi Owa and others. Closely related to Hankel determinants are

the Toepliz determinants. A Toeplitz matrix Tq(n) defined by

Tq(n) :=

∣∣∣∣∣∣∣∣∣∣∣∣

an an+1 · · · an+q−1

an+1 an · · ·...

... · · ·...

an+q−1 an+q · · · an

∣∣∣∣∣∣∣∣∣∣∣∣

A Toeplitz matrix can be thought of as an upside-down Hankel matrix, in that Hankel ma-

trices have constant entries along the reverse diagonal, whereas Toeplitz matrices have constant

entries along the diagonal. A good summary of the applications of Toeplitz matrices to a wide

range of areas of pure and applied mathematics can also be found in [11].

We aim to define q-starlike, q-convex functions and Ma-Minda starlike and convex func-

tions. We use the concept of principle of subordination and q-calculus to define our classes.

Recently in the second half of the twentieth century q-calculus aroused interest due to lot of

applications in the various mathematical fields such as combinatorics, number theory, quantum

theory and the theory of relativity. The q-derivative of a function is defined in the following.

Definition 1.1([9]) The q-derivative of f is given by

∂qf(z) =

f(z)−f(qz)z(1−q) , z 6= 0,

f ′(0), z = 0., where 0 < q < 1. (1.2)

Equivalently, (1.2) may be written as

∂qf(z) = 1 +

∞∑

n=2

[n]qanzn−1, z 6= 0

where

[n]q =

1−qn

1−q, q 6= 1

n, q= 1.

72 Hamid Shamsan, Fuad S. M. Alsarari and S. Latha

Note that as q → 1−, [n]q → n.

Definition 1.2([7]) Let f be analytic in U and be given by (1.1). Then a function f is starlike

if and only if, ℜ{

zf ′(z)f(z)

}> 0. We denote the class of starlike functions by S∗.

The class of functions with positive real part plays a significant role in complex function

theory. Using principle of subordination we define the functions with positive real part.

Definition 1.3([17]) Let f and g be analytic in U , then f is said to be subordinate to the

function g, written f(z) ≺ g(z), if there exists an analytic function ω : U → U satisfying

ω(0) = 0 and |ω(z)| < 1 such that f(z) = g(ω(z)), z ∈ U .

Definition 1.4([3]) Let P denote the class of analytic functions p : U → C, p(0) = 1,

and ℜ{p(z)} > 0, then p(z) ≺ 1+z1−z

.

The class P can be completely characterized in terms of subordination. We need the

following lemmas to derive our results.

Lemma 1.5([3]) If the function p ∈ P is given by the series

p(z) = 1 + c1z + c2z2 + c3z

3 + · · · , (1.3)

then the following sharp estimate holds:

|cn| ≤ 2 (n = 1, 2, · · · ).

Lemma 1.6([6]) If the function p ∈ P is given by the series (1.3), then

2c2 = c21 + x(4 − c21), (1.4)

4c3 = c31 + 2(4 − c21)c1x− c1(4 − c21)x2 + 2(4 − c21)(1 − |x|2)z, (1.5)

for some x, z with |x| ≤ 1 and |z| ≤ 1.

§2. Main Results

Definition 2.1 Let ϕ : U → C be analytic, and let the Maclaurin series of ϕ is given by

ϕ(z) = 1 +B1z +B2z2 +B3z

3 + · · · (B1, B2 ∈ R, B1 > 0). (2.1)

Let 0 ≤ γ ≤ 1 and τ ∈ C\{0}. A function f ∈ A is in the class Rτq,γ(ϕ) if it satisfies the

following subordination:

1 +1

τ(∂qf(z) + γz∂2

qf(z) − 1) ≺ ϕ(z).


Theorem 2.2 Let 0 ≤ γ ≤ 1, τ ∈ C\{0} and let the function f as in (1.1) be in the class

Rτq,γ(ϕ). Also let p =

[2]q[4]q(1+γ)(1+[3]qγ)([3]q)2(1+[2]qγ)2 .

(1) If B1, B2 and B3 satisfy the conditions 2|B2|(1− p) +B1(1− 2p) ≤ 0, |B1B3 − pB22 | −

pB21 ≤ 0, then the second Hankel determinant satisfies

|a2a4 − a23| ≤

|τ |2B21

([3]q)2(1 + [2]qγ)2.

(2) If B1, B2 and B3 satisfy the conditions 2|B2|(1− p)+B1(1− 2p) ≥ 0, 2|B1B3 − pB22 |−

2(1−p)B1|B2|−B1 ≥ 0, or the conditions 2|B2|(1−p)+B1(1−2p) ≤ 0, |B1B3−pB22 |−pB2

1 ≥0, then the second Hankel determinant satisfies

|a2a4 − a23| ≤

|τ |2[2]q[4]q(1 + γ)(1 + [3]qγ)

|B1B3 − pB22 |.

(3) If B1, B2 and B3 satisfy the conditions 2|B2|(1− p) +B1(1− 2p) > 0, |B1B3 − pB22 | −

pB21 ≤ 0, then the second Hankel determinant satisfies

|a2a4 − a23| ≤

|τ |2B21

4[4]q[2]q(1 + γ)(1 + [3]qγ)

×[4p|B1B3 − pB2

2 | − 4B1(1 − p)[|B2|(3 − 2p) +B1] − 4B22(1 − p)2 −B2

1(1 − 2p)2

|B1B3 − pB22 | −B1(1 − p)(2|B2| +B1)

].

Proof Since f ∈ Rτq,γ(ϕ), there exists an analytic function w with w(0) = 0 and |w(z)| < 1

in U such that

1 +1


qf(z) − 1) = ϕ(w(z)). (2.2)

Define the function p1 by

p1(z) =1 + w(z)

1 − w(z)= 1 + c1z + c2z

2 + · · · ,

or equivalently,

w(z) =p1(z) − 1

p1(z) + 1=

1

2

(c1z +

(c2 −

c212

)z2 +

(c3 − c1c2 +

c314

)z3 + · · ·

)(2.3)

Then p1 is analytic in U with p1(0) = 0 and has a positive real part in U . By using (2.3)

together with (2.1), it is evident that

ϕ

(p1(z) − 1

p1(z) + 1

)= 1 +

1

2B1c1z +

(1

2B1

(c2 −

c212

)+

1

4B2c

21

)z2 + · · · (2.4)


since f has the Maclaurin series given by (1.1), a computation shows that

1 +1


qf(z) − 1) = 1 +[2]qa2(1 + γ)

τz +

[3]qa3(1 + [2]qγ)

τz2

+[4]qa4(1 + [3]qγ)

τz3 + · · · . (2.5)

It follows from (2.2), (2.4) and (2.5) that

a2 =τB1c1

2[2]q(1 + γ),

a3 =τB1

4[3]q(1 + [2]qγ)

[2c2 + c21(

B2

B1− 1)

],

a4 =τ

8[4]q(1 + [3]qγ)

[B1(4c3 − 4c1c2 + c31) + 2B2c1(2c2 − c21) +B3c

31

].

Therefore,

a2a4 − a23

=τ2B1c1

16([4]q[2]q)(1 + γ)(1 + [3]qγ)

[B1(4c3 − 4c1c2 + c31) + 2B2c1(2c2 − c21) +B3c

31

]

− τ2B21

16([3]q)2(1 + [2]qγ)2

[4c22 + c41(

B2

B1− 1)2 + 4c2c

21(B2

B1− 1)

]

=τ2B1c1

16([4]q[2]q)(1 + γ)(1 + [3]qγ)

{[(4c1c3 − 4c2c

21 + c41

)+

2B2c21

B1

(2c2 − c21

)+B3

B1c41

]

− [4]q[2]q(1 + γ)(1 + [3]qγ)

([3]q)2(1 + [2]qγ)2

[4c22 + c41(

B2

B1− 1)2 + 4c2c

21(B2

B1− 1)

]},

which yields

|a2a4 − a23| = T

∣∣∣∣4c1c3 + c41

[1 − 2

B2

B1− p(

B2

B1− 1)2 +

B3

B1

]

−4pc22 − 4c21c2

[1 − B2

B1+ p(

B2

B1− 1)

]∣∣∣∣ , (2.6)

where,

T =|τ |2B2

1

16([4]q[2]q)(1 + γ)(1 + [3]qγ)and p =

[4]q[2]q(1 + γ)(1 + [3]qγ)

([3]q)2(1 + [2]qγ)2.

It can be easily verified that p ∈ [6481 ,89 ] for 0 ≤ γ ≤ 1 and 0 ≤ q ≤ 1. Let

d1 = 4, d2 = −4[1 − B2

B1+ p(B2

B1− 1)

],

d3 = −4p, d4 =[1 − 2B2

B1− p(B2

B1− 1)2 + B3

B1

].

(2.7)

Then (2.6) becomes

∣∣a2a4 − a23

∣∣ = T∣∣d1c1c3 + d2c

21c2 + d3c

22 + d4c

41

∣∣ . (2.8)


Since the function p(eiθz)(θ ∈ R) is in the class P for any p ∈ P , there is no loss of

generality in assuming c1 > 0. Write c1 = c, c ∈ [0, 2]. Substituting the values of c2 and c3

respectively from (1.6) and (1.5) in (2.8), we obtain

|a2a4 − a23| =

T

4

∣∣c4 (d1 + 2d2 + d3 + 4d4) + 2xc2(4 − c2) (d1 + d2 + d3)

+ (4 − c2)x2(−d1c

2 + d3(4 − c2))

+ 2d1c(4 − c2)(1 − |x|2z)∣∣ .

Replacing |x| by µ and substituting the values of d1, d2, d3 and d4 from (2.7) yields

|a2a4 − a23| ≤ T

4

[4c4∣∣∣∣B3

B1− p

B22

B21

∣∣∣∣+ 8

∣∣∣∣B2

B1

∣∣∣∣µc2(4 − c2)(1 − p)

+(4 − c2)µ2(4c2 + 4p(4 − c2)) + 8c(4 − c2)(1 − µ2)]

= T

[c4∣∣∣∣B3

B1− p

B22

B21

∣∣∣∣+ 2c(4 − c2) + 2µ

∣∣∣∣B2

B1

∣∣∣∣ c2(4 − c2)(1 − p)

+µ2(4 − c2)(1 − p)(c− α)(c − β)]≡ F (c, µ), (2.9)

where α = 2, β = 2p/(1 − p) > 2.

Note that for (c, µ) ∈ [0, 2]× [0, 1], differentiating F (c, µ) in (2.9) partially with respect to

µ yields∂F

∂µ= T

[2

∣∣∣∣B2

B1

∣∣∣∣ c2(4 − c2)(1 − p) + 2µ(4 − c2)(1 − p)(c− α)(c− β)

]. (2.10)

Then, for 0 < µ < 1, 0 < q < 1 and any fixed c with 0 < c < 2, it is clear from (2.10) that∂F∂µ

> 0, that is, F (c, µ) is an increasing function of µ. Hence, for fixed c ∈ [0, 2], the maximum

of F (c, µ) occurs at µ = 1, and

maxF (c, µ) = F (c, 1) ≡ G(c),

which is

G(c) = T

{c4[∣∣∣∣B3

B1− p

B22

B21

∣∣∣∣− (1 − p)

(2

∣∣∣∣B2

B1

∣∣∣∣+ 1

)]+ 4c2

[2

∣∣∣∣B2

B1

∣∣∣∣ (1 − p) + 1 − 2p

]+ 16p

}.

Let

X =

∣∣∣∣B3

B1− p

B22

B21

∣∣∣∣− (1 − p)

(2

∣∣∣∣B2

B1

∣∣∣∣+ 1

),

Y = 4

[2

∣∣∣∣B2

B1

∣∣∣∣ (1 − p) + 1 − 2p

], (2.11)

Z = 16p.

Since

max(Xt2 + Y t+ Z) =

Z, Y ≤ 0, X ≤ −Y4 ;

16X + 4Y + Z, Y ≥ 0, X ≥ −Y8 orY ≤ 0, X ≥ −Y

4 ;

4XZ−Y 2

4X, Y > 0, X ≤ −y

8 ,

(2.12)


where 0 ≤ t ≤ 4. Then we have

|a2a4 − a23| ≤ B1

16([4]q − 1)([3]q − 1)([2]q − 1)

×

Z, Y ≤ 0, X ≤ −Y4 ;

16X + 4Y + Z, Y ≥ 0, X ≥ −Y8 or Y ≤ 0, X ≥ −Y

4 ;

4XZ−Y 2

4X, Y > 0, X ≤ −y

8 ,

where X,Y and Z are given by (2.11). 2Remark 2.3 notice that

(1) As q → 1− Theorem 2.2 reduces to Theorem 3 in [12].

(2) As q → 1− for the choice ϕ(z) := (1 + Az)/(1 + Bz) with −1 ≤ B < A ≤ 1 Theorem

2.2 reduces to Theorem 2.1 in [12].

Definition 2.4 An analytic function f is close-to-q-convex in U , if and only if, there exists

g ∈ S∗q such that

ℜ{z∂qf(z)

g(z)

}> 0.

We denote the class of close-to-q-convex functions by Kq.

For f ∈ S∗, we can write z∂qf(z) = f(z)h(z), where h ∈ P , the class of function satisfying

ℜh(z) > 0 for z ∈ U and

h(z) = 1 +

∞∑

n=2

cnzn.

For f ∈ Kq, we can write z∂qf(z) = g(z)p(z), where p ∈ P and

p(z) = 1 +

∞∑

n=2

pnzn.

Theorem 2.5 Let f ∈ Kq and given by (1.1) with associated starlike function g define by

g(z) = z +

∞∑

n=2

bnzn.

Then

T2(2) = |a23 − a2

2| ≤ [5]q, (b2 ∈ R)

and the inequality is sharp.

Proof Write z∂qf(z) = g(z)h(z) and zg′(z) = g(z)p(z), with

h(z) = 1 +∞∑

n=1

cnzn


and

p(z) = 1 +

∞∑

n=1

pnzn.

Then equating the coefficients in z∂qf(z) = g(z)h(z) where coefficients’ relations from zg′(z) =

g(z)p(z) is also used, we obtain

a2 =c1 + p1

[2]q,

a3 =p21 + p2 + 2p1c1 + 2c2

2[3]q

so that

|a23 − a2

2| =

∣∣∣∣−1

[2]2qc21 +

1

[3]2qc22 −

2

[2]2qc1p1 +

2

[3]2qc1c2p1 −

1

[2]2qp21 +

1

[3]2qc21p

21 +

1

[3]2qc2p

21

+1

[3]2qc1p

31 +

1

4[3]2qp41 +

1

[3]2qc2p2 +

1

[3]2qc1p1p2 +

1

2[3]2qp21p2 +

1

4[3]2qp22

∣∣∣∣ .

We now use Lemma 1.6 to express c2 and p2 in terms of c1 and p1 and writingX = 4 − c21and Y = 4 − p2

1 for simplicity to get

|a23 − a2

2|

=

∣∣∣∣−1

[2]2qc21 +

1

4[3]2qc41 −

2

[2]2qc1p1 +

1

[3]2qc31p1 −

1

[2]2qp21 +

7

4[3]2qc21p

21

+3

2[3]2qc1p

31 +

9

16[3]2qp41 +

1

2[3]2qc21xX +

1

[3]2qc1p1xX +

3

4[3]2qp21xX

+1

4[3]2qx2X2 +

1

4[3]2qc21yY +

1

2[3]2qc1p1yY

+3

8[3]2qp21yY +

1

4[3]2qxXyY +

1

16[3]2qy2Y 2

∣∣∣∣ .

Without loss in generality we can assume that c1 = c where 0 ≤ c ≤ 2. Also since we are

assuming b2 = p1 to be real, we can write p1 = r, with 0 ≤ |r| ≤ 2, and write |r| = p. We

note at this point a further normalisation of p1 to be real would remove the requirement that

p1 = b2 is real, but such normalisation does not appear to be justified. It follows from Lemma

1.6 that with now X = 4 − c2 and Y = 4 − p2. So,

|a23 − a2

2|

≤∣∣∣∣−1

[2]2qc2 +

1

4[3]2qc4 − 2

[2]2qcp+

1

[3]2qc3p− 1

[2]2qp2 +

7

4[3]2qc2p2 +

3

2[3]2qcp3 +

9

16[3]2qp4

∣∣∣∣

+1

2[3]2qc2|x|X +

1

[3]2qcp|x|X +

3

4[3]2qp2|x|X +

1

4[3]2q|x|2X2 +

1

4[3]2qc2|y|Y

+1

2[3]2qcp|y|Y +

3

8[3]2qp2|y|Y +

1

4[3]2q|x|X |y|Y +

1

16[3]2q|y|2Y 2.


Now we assume |x| ≤ 1 and |y| ≤ 1 and simplify to obtain

|a23 − a2

2| ≤∣∣∣∣−1

[2]2qc2 +

1

4[3]2qc4 − 2

[2]2qcp+

1

[3]2qc3p− 1

[2]2qp2 +

7

4[3]2qc2p2 +

3

2[3]2qcp3 +

9

16[3]2qp4

∣∣∣∣

+9

[3]2q− 1

4[3]2qc4 +

6

[3]2qcp− 1

[3]2qc3p+

3

[3]2qp2 − 3

4[3]2qc2p2 − 1

2[3]2qcp3 − 5

16[3]2qp4.

Suppose that the expression between the modulus signs is positive, then

|a23 − a2

2| ≤ ψ1(c, p) =9

[3]2q− 1

[2]2qc2 +

2(3[2]2q − [3]2q)

[2]2q[3]2qcp

+2(3[2]2q − [3]2q)

[2]2q[3]2qp2 +

1

[3]2qc2P 2 +

1

[3]2qcp3 +

1

4[3]2qp4.

Then for 0 ≤ c ≤ 2 and 0 ≤ p ≤ 2 and fixed q with 0 < q < 1 and calculus we get that ψ1(c, p)

has a maximum value of [5]q at [0,2].

If the expression between the modulus signs is negative, then

|a23 − a2

2| ≤ ψ2(c, p) =9

[3]2q+

1

[2]2qc2 − 1

2[3]2qc4 +

2(3[2]2q + [3]2q)

[2]2q[3]2qcp

− 2

[3]2qc3p

3[2]2q + [3]2q[2]2q[3]2q

p2 − 5

2[3]2qc2P 2 − 2

[3]2qcp3 − 7

8[3]2qp4.

Then for 0 ≤ c ≤ 2 and 0 ≤ p ≤ 2 and fixed q with 0 < q < 1 and calculus we get that ψ2(c, p)

has a maximum value less than [3]q . Thus the proof is complete. 2As q → 1−, we have following result due to D. K. Thomas and S. Abdul Halim [18].

Corollary 2.6 Let f ∈ K and be given by (??) with the associated starlike function g be defined

by

g(z) = z +

∞∑

n=2

bnzn.

Then

T2(2) = |a23 − a2

2| ≤ 5,

provided b2 is real. The inequality is sharp.

References

[1] H. S. Al-Amiri and M. O. Reade, On a linear combination of some expressions in the theory

of the univalent functions, Monatshefte Math., 80(4)(1975), 257-264.

[2] D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions,

Appl. Math. Lett., 26(1)(2013), 103-107.

[3] P. L. Duren, Univalent Functions. Grundlehren der Mathematischen Wissenschaften,

Springer, New York, 259(1983).


[4] M. Fekete and G. Szego , Eine Bermerkung Uber ungerade schlichte Funktionen, J. London

Math. Soc., 1(2)(1933), 85-89.

[5] S. Gagandeep and S. Gurcharanjit, Second Hankel determinant for a Subclass of alpha

Convex functions, J. Appl.Computat Math., 3(2014), 1-3.

[6] U. Grenander, G. Szego, Toeplitz Forms and Their Applications, California Monographs

in Mathematical Sciences. University of California Press, Berkeley, (1958).

[7] A. W. Goodman, Univalent Functions, Vol I, II. Washington, New Jersey: Polygonal

Publishing House, (1983).

[8] W. K. Hayman, On the second Hankel determinant of mean univalent functions, Proc.

London Math. Soc., 3(18)(1968), 77-94.

[9] F. H. Jackson,On q-functions and a certain difference operator, Trans. Royal Soc. Edin-

burgh, 46(1909), 253-281.

[10] A. Janteng, S. Halim and M. Darus, Hankel determinant forstarlike and convex functions,

Int. J. Math. Anal. (Ruse), 1 (2007), 619-625.

[11] Y. Ke and Lek-Heng Lim, Every matrix is a product of Toeplitz matrices, Foundations of

Computational Mathematics, 16(30) (2016), 577-598.

[12] L. S. Keong, V. Ravichandran, and S. Supramaniam. Bounds for the second Hankel de-

terminant of certain univalent functions, arXiv :1303.0134 (2013).

[13] M. S. Liu, J. F. Xu and M. Yang, Upper bound of second Hankel determinant for certain

subclasses of analytic functions, Abstr. Appl. Anal., 2014.

[14] K. I. Noor and S. A. Al-Bany, On Bazilevic functions, Internat. J. Math. Math. Sci.,

1(10)(1987), 79-88.

[15] C. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J.

London Math. Soc., 41(1966), 111-122.

[16] C. Pommerenke, On the Hankel determinants of univalent functions, Math., 14(1967),

108-112.

[17] C. Pommerenke, G. Jensen, Univalent functions, Vandenhoeck und Ruprecht, 25, 1975.

[18] D. K. Thomas, S. A. Halim, Toeplitz matrices whose elements are the coefficients of starlike

and close-to-convex functions, Bulletin of the Malaysian Mathematical Sciences Society,

(2016), 1-10.


On Obic Algebras

Ilojide Emmanuel

(Department of Mathematics, Federal University of Agriculture, Abeokuta 110101, Nigeria)


Abstract: In this paper, obic algebras are introduced and their properties are investigated.

Homomorphisms and krib maps as well as monics of obic algebras are studied. Properties

of implicative obic algeras are also investigated.

Key Words: Obic algebras, krib maps, monics.

AMS(2010): 20N02, 20N05, 06F35.

§1. Introduction

Algebras of type (2,0) are well known types of algebraic structures. They comprise non-empty

sets, some constant element together with a binary operation. In [1], Kim and Kim introduced

the notion of BE-algebras. Ahn and so, in [2] and [3] introduced the notions of ideals and upper

sets in BE-algebras and investigated related properties. In this paper, a new class of algebras

called obic algebras are introduced. Their properties are investigated. Homomorphisms and

krib maps as well as monics of obic algebras are studied. Moreover, translations in obic algebras

are investigated as well as properties of implicative obic algebras.

Definition 1.1 A non-empty set X together with a binary operation ∗ defined on X is called

a groupoid.

Definition 1.2 A triple (X ; ∗, 0), where X is a non-empty set, ∗ a binary operation on X and 0

a constant element of X is called an obic algebra if the following axioms hold for all x, y, z ∈ X:

(1) x ∗ 0 = x;

(2) [x ∗ (y ∗ z)] ∗ x = x ∗ [y ∗ (z ∗ x)];(3) x ∗ x = 0.

Example 1.1 Consider the multiplicative group G = {1,−1, i,−i}. Define a binary operation

∗ on G by a ∗ b = ab−1. Then (G; ∗, 1) is an obic algebra.

Example 1.2 Let Z denote the set of integers. Then (Z;−, 0) is an obic algebra.

Example 1.3 Let X = {0, 1}. Define a binary operation ∗ on X in Table 1.

1Received June 11, 2019, Accepted December 2, 2019.

On Obic Algebras 81

∗ 0 1

0 0 1

1 1 0

Table 1

Then, (X, ∗, 0) is an obic algebra.

We shall adopt the notation X for an obic algebra (X ; ∗, 0) unless stated otherwise.

Definition 1.3 An obic algebra is called simple if y ∗ (z ∗ x) = x ∗ (y ∗ z) for all x, y, z ∈ X.

Definition 1.4 An obic algebra is called plain if 0 ∗ (y ∗ z) = (0 ∗ y) ∗ z for all y, z ∈ X.

Definition 1.5 An obic algebra X is said to have the weak property (WP) if x ∗ y = 0 and

y ∗ x = 0 imply that x = y.

Definition 1.6 An obic algebra X is called prime if 0 ∗ x = 0 for all x ∈ X.

Lemma 1.1 Let X be an obic algebra. Then for all x, y ∈ X, the following hold:

x ∗ y = [x ∗ (y ∗ x)] ∗ x.

Definition 1.7 A non-empty subset S of an obic algebra X is called a subalgebra if S is an

obic algebra with respect to the binary operation in X.

Example 1.4 Let X be an obic algebra. Then X and {0} are subalgebras of X .

Example 1.5 LetX be the obic algebra in example 1.1. Then the subset {1,−1} is a subalgebra

of X .

The following results are immediately obtained by the definition.

Proposition 1.1 A non-empty subset S of an obic algebra is a subalgebra if and only if the

following hold:

(1) 0 ∈ S;

(2) x ∗ y ∈ S for all x, y ∈ S.

Proposition 1.2 Let X be a plain obic algebra. Then, the subset S = {x ∈ X : 0 ∗ x = 0} is a

subalgebra of X.

§2. Obic Homomorphisms

Definition 2.1 Let (X ; ∗, 0) and (Y ; ◦, 0′) be obic algebras. A function f : X → Y is called

an obic homomorphism if f(a ∗ b) = f(a) ◦ f(b) for all a, b ∈ X.

Definition 2.2 Let f : X → Y be an obic homomorphism. The set {x ∈ X : f(x) = 0′} is

called the kernel of f .

82 Ilojide Emmanuel

Proposition 2.1 Let f : X → Y be an obic homomorphism. Then the kernel of f is a

subalgebra of X.

Then, we get conclusions following by definition.

Proposition 2.2 Let f : X → Y be an obic homomorphism. Then,

(1) f(0) = 0′;

(2) x ∗ y = 0 ⇒ f(x) ◦ f(y) = 0′ for all x, y ∈ X.

Let f : X → Y be an obic homomorphism. Define a relation ∼ by (x ∼ y) ⇔ f(x) = f(y).

Then, we know

Lemma 2.1 Let f : X → Y be an obic homomorphism. The relation ∼ defined by (x ∼ y) ⇔f(x) = f(y) is an equivalence relation.

Definition 2.3 An equivalence relation ∼ on an obic algebra X is called a congruence if (x ∼ y)

and (u ∼ v) ⇒ (x ∗ u) ∼ (y ∗ v).

We have the following result by definition.

Lemma 2.2 Let f : X → Y be an obic homomorphism. The equivalence relation ∼ defined by

(x ∼ y) ⇒ f(x) = f(y) is a congruence.

Let [x] be the equivalence class of x ∈ X and let X denote the collection of equivalence

classes in the equivalence relation ∼. Define a binary operation ⋄ on X by [x] ⋄ [y] = [x ∗ y].

Theorem 2.1 Let f : X → Y be an obic homomorphism. Then (X ; ⋄, [0]) is an obic algebra.

Proof By Lemma 2.2, the binary operation ⋄ is well-defined. Now, let [x], [y], [z] ∈ X.

Consider [x] ⋄ [0] = [x ∗ 0] = [x]. Also,

([x] ⋄ ([y] ⋄ [z])) ⋄ [x] = ([x] ⋄ [y ∗ z]) ⋄ [x] = ([x ∗ (y ∗ z)]) ⋄ [x]

= [(x ∗ (y ∗ z)) ∗ x] = [x ∗ (y ∗ (z ∗ x))]= [x] ⋄ ([y] ⋄ ([z] ⋄ [x])).

Also, [x] ⋄ [x] = [x ∗ x] = [0]. 2Theorem 2.2 Let f : X → X be an endomorphism. Then f(X) is isomorphic to X.

Proof Consider the map φ : f(X) → X such that φ(y) = [y]. Let y1, y2 ∈ f(X). Then

φ(y1 ∗ y2) = [y1 ∗ y2] = [y1] ⋄ [y2] = φ(y1) ⋄ φ(y2). Also, φ is one to one and onto. 2Theorem 2.3 Let φ : X → X be an obic homomorphism; where X has the weak property.

Then φ is one to one if and only if ker(φ) = {0}.

Proof Suppose φ is one to one. Let x ∈ ker(φ). Then φ(x) = 0 = φ(0). So, ker(φ) = {0}.

On Obic Algebras 83

Conversely, suppose ker(φ) = {0}. Let x, y ∈ X such that φ(x) = φ(y). Then φ(x ∗ y) =

φ(x) ∗ φ(y) = 0. Also, φ(y ∗ x) = 0. So, (x ∗ y), (y ∗ x) ∈ ker(φ). Hence φ is one to one. 2Definition 2.4 An obic homomorphism f : X → X is called idempotent if f(f(x)) = f(x) for

all x ∈ X.

Theorem 2.4 Let X be an obic algebra with weak property. Let φ be an idempotent endomor-

phism on X. Then φ is one to one if and only if φ is the identity map.

Proof Suppose φ is one to one. Let x ∈ X . Then φ((x ∗ φ(x))) = φ(x) ∗ φ(φ(x)) =

φ(x) ∗ φ(x) = 0 = φ(0).So, x ∗ φ(x) = 0. Similarly argument gives φ(x) ∗ x = 0. And so

φ(x) = x. Hence φ is the identity map.

The converse is obvious. 2§3. Implicative Obic Algebras

Definition 3.1 An obic algebra X is called implicative if x ∗ (y ∗ x) = x for all x, y ∈ X.

The following conclusion can be obtained by the definition.

Lemma 3.1 Let X be an implicative obic algebra. Then the following hold:

(1) 0 ∗ 0 = 0;

(2) x ∗ y = (x ∗ y) ∗ (0 ∗ y);(3) x ∗ y = (x ∗ (y ∗ x)) ∗ y;(4) y ∗ x = y ∗ (x ∗ (y ∗ x)).

Definition 3.2 Let X be an obic algebra. Let x be a fixed element of X. The map Lx : X → X

such that Lx(a) = x ∗ a for all a ∈ X is called a left translation on X. Similarly, the map

Rx : X → X such that Rx(a) = a ∗ x for all a ∈ X is called a right translation on X.

Theorem 3.1 Let Lx : X → X be an endomorphism. Then x = 0. Moreover, if X is

implicative, then x = x ∗ (x ∗ y).

Proof Consider x = x ∗ 0 = Lx(0) = Lx(0 ∗ 0) = Lx(0) ∗Lx(0) = (x ∗ 0) ∗ (x ∗ 0) = 0. Now

suppose X is implicative. Let y ∈ X . Then x = x ∗ 0 = Lx(0) = Lx(0 ∗ y) = Lx(0) ∗ Lx(y) =

x ∗ (x ∗ y). 2Denote the collection of left translations on an obic algebra X by L(X) and define a binary

operation ⊙ on L(X) by (La ⊙ Lb)(x) = La(x) ∗ Lb(x) for all x ∈ X .

Theorem 3.2 Let X be an implicative obic algebra. Then (L(X);⊙, L0) is an obic algebra.

Proof Let La, Lb, Lc ∈ L(X). For every x ∈ X , consider (La ⊙ L0)(x) = La(x) ∗ L0(x) =

(a ∗ x) ∗ (0 ∗ x) = (a ∗ x) = La(x). So, La ⊙ L0 = La.

84 Ilojide Emmanuel

Also consider

(La ⊙ (Lb ⊙ Lc) ⊙ La)(x) = (a ∗ x) ∗ ((b ∗ x) ∗ ((c ∗ x) ∗ (a ∗ x)))= (La ⊙ (Lb ⊙ (Lc ⊙ La)))(x).

So,

(La ⊙ (Lb ⊙ Lc) ⊙ La) = (La ⊙ (Lb ⊙ (Lc ⊙ La))).

And clearly,

(La ⊙ La)(x) = La(x) ∗ La(x) = (a ∗ x) ∗ (a ∗ x) = 0 = L0(x). 2Corollary 3.1 Let X be a prime obic algebra. Then (L(X);⊙, L0) is an obic algebra.

Corollary 3.2 (L(X);⊙, L0) is prime if and only if X is prime.

We therefore know that

Proposition 3.1 Let X be an obic algebra. Then the translation L0 : X → X commutes with

any endomorphism on X.

Definition 3.3 An obic algebra X is said to have the distributive property if 0 ∗ (x ∗ y) =

(0 ∗ x) ∗ (0 ∗ y) for all x, y ∈ X.

Proposition 3.2 Let X be an obic algebra with distributive property. Then the translation

L0 : X → X is the only homomorphism in the collection L(X).

Proof Clearly, L0 is a homomorphism. Let x ∈ X such that x 6= 0. Let Suppose Lx is a

homomorphism on X . Consider x = (x ∗ 0) = Lx(0) = Lx(0 ∗ 0) = Lx(0) ∗ Lx(0) = 0; which is

a contradiction. 2§4. Krib Maps in Obic Algebras

Definition 4.1 Let X be an obic algebra. A self map α : X → X is called a right krib map if

α(x ∗ y) = x ∗ α(y) for all x, y ∈ X.

If α(x ∗ y) = α(x) ∗ y for all x, y ∈ X, then α is called a left krib map. α is called a krib

map if it is both a right and a left krib map.

Example 4.1 Consider the obic algebra X in Example 1.3. Define α : X → X by α(1) =

0, α(0) = 1. Then α is a right krib map of X .

Denote by D(X) the collection of right krib maps on an obic algebra X . We know the next

result by definition.

Proposition 4.1 Let X be an obic algebra. Then D(X) is a monoid.

On Obic Algebras 85

Definition 4.2 Let X be an obic algebra. A map α : X → X is called regular if α(0) = 0. If

α(0) 6= 0, then α is called irregular.

The following results can be verified immediately.

Lemma 4.1 Let α be an irregular right krib map of an obic algebra X. Then the following

hold for all x ∈ X:

(1) x ∗ α(x) 6= 0;

(2) α(x) = x ∗ α(0).

Proposition 4.2 Every right krib map of a prime obic algebra is regular.

Corollary 4.1 Let α be a krib map on a prime obic algebra X. Then α is regular.

Proposition 4.3 A right krib map α of an obic algebra X is regular if and only if x ∗α(x) = 0

for all x ∈ X.

Proof Suppose α is regular. Then 0 = α(0) = xα(x). Conversely, suppose x ∗ α(x) = 0.

Then α(0) = x ∗ α(x) = 0. 2Proposition 4.4 A left krib map α of an obic algebra X is regular if and only if α(x) ∗ x = 0

for all x ∈ X.

Theorem 4.1 Let α be a krib map of an obic algebra X. Then the following are equivalent:

(1) x ∗ α(x) = 0;

(2) α is regular;

(3) α(x) ∗ x = 0.

Proof The proof is straightforward by definition. 2§5. Monics of Obic Algebras

Let X be an obic algebra. Define ’∧’ by x ∧ y = y ∗ (y ∗ x) for all x, y ∈ X .

Definition 5.1 Let X be an obic algebra. A function θ : X → X is called a left (resp.

right)monic if θ(x ∗ y) = (θ(x) ∗ y) ∧ (x ∗ θ(y)) (resp. θ(x ∗ y) = (x ∗ θ(y)) ∧ (θ(x) ∗ y)) for all

x, y ∈ X.

If θ : X → X is both a left and a right monic, then θ is called a monic.

Example 5.1 Let X be the obic algebra given by Table 2.

∗ 0 1

0 0 1

1 1 0

Table 2

86 Ilojide Emmanuel

The map θ : X → X such that θ(1) = 0, θ(0) = 1 is a left monic.

Definition 5.2 Let X be an obic algebra. A map α : X → X is called regular if α(0) = 0.

Definition 5.3 Let X be an obic algebra. A self map θ on X is called self preserving if

θ(x) ∗ x = x for all x ∈ X.

Definition 5.4 Let X be an obic algebra. A self map θ on X is called anti-self preserving if

x ∗ θ(x) = x for all x ∈ X.

Definition 5.5 Let X be an obic algebra. A self map θ on X is called preserving if it is both

self-preserving and ant-self-preserving.

Proposition 5.1 Let θ be a regular left monic on an obic algebra X. Then,

(x ∗ θ(x)) ∗ [(x ∗ θ(x)) ∗ (θ(x) ∗ x)] = (y ∗ θ(y)) ∗ [(y ∗ θ(y)) ∗ (θ(y) ∗ y)]

for all x, y ∈ X.

Proof Now, 0 = θ(0) = θ(x ∗ x) = (x ∗ θ(x)) ∗ [(x ∗ θ(x)) ∗ (θ(x) ∗ x)]. Similar argument

gives also (y ∗ θ(y)) ∗ [(y ∗ θ(y)) ∗ (θ(y) ∗ y)] = 0. Hence, the conclusion follows. 2Proposition 5.2 Let X be a regular left monic on an associative obic algebra X. Then

0 ∗ [θ(x) ∗ x] = 0 ∗ [θ(y) ∗ y] for all x, y ∈ X.

Proof By proposition 5.1,

0 = [x ∗ θ(x)] ∗ [(x ∗ θ(x)) ∗ (θ(x) ∗ x)]= [(x ∗ θ(x)) ∗ (x ∗ θ(x))] ∗ [θ(x) ∗ x]= 0 ∗ [θ(x) ∗ x].

Similarly, we have 0 ∗ [θ(y) ∗ y] = 0. The conclusion follows. 2Proposition 5.3 Let θ be a self preserving left monic on an obic algebra X. Then

[x ∗ θ(x)] ∗ [(x ∗ θ(x)) ∗ x)] = θ(0) for all x ∈ X.

Proof Now,

θ(0) = θ(x ∗ x)= [θ(x) ∗ x] ∧ [x ∗ θ(x)]= [x ∗ θ(x))] ∗ [(x ∗ θ(x)) ∗ x]. 2

Corollary 5.1 Let θ be a regular self preserving left monic on an obic algebra X.

Then [x ∗ θ(x))] ∗ [(x ∗ θ(x)) ∗ x] = 0 for all x ∈ X.

The following propositions can be immediately verified.

On Obic Algebras 87

Proposition 5.4 Let θ be a regular self preserving left monic on an obic algebra X. Then

[x ∗ θ(x))] ∗ [(x ∗ θ(x)) ∗ x] = [y ∗ θ(y))] ∗ [(y ∗ θ(y)) ∗ y] for all x, y ∈ X.

Proposition 5.5 Let θ be a self preserving left monic on an associative obic algebra X. Then

0 ∗ x = θ(0) for all x ∈ X.

Proposition 5.6 Let θ be a regular self preserving left monic on an associative obic algebra

X. Then 0 ∗ [θ(x) ∗ x] = 0 ∗ [θ(y) ∗ y] for all x, y ∈ X.

Proposition 5.7 Let θ be a regular self preserving left monic on an associative obic algebra

X. Then 0 ∗ x = 0 for all x ∈ X.

Theorem 5.1 Let X be an associative obic algebra with a self preserving left monic θ. Then

X is prime if and only if θ is regular.

Proof Suppose X is prime. Then,

0 = 0 ∗ x = [(x ∗ θ(x)) ∗ (x ∗ θ(x))] ∗ x= (x ∗ θ(x)) ∗ [(x ∗ θ(x)) ∗ x]= x ∧ [x ∗ θ(x)]= θ(x ∗ x) = θ(0).

Conversely, suppose θ is regular. Then,

0 = θ(0) = θ(x ∗ x) = [(θ(x) ∗ x)] ∧ [x ∗ θ(x)]= x ∧ [x ∗ θ(x)] = 0 ∗ x. 2

The two conclusions following can be easily verified by definition.

Proposition 5.8 Let θ be an anti-self preserving left monic on an obic algebra X. Then

x ∗ [x ∗ θ(x)] = θ(0) for all x ∈ X.

Proposition 5.9 Let θ be a regular anti-self preserving left monic on an obic algebra X. Then

y ∗ [y ∗ θ(y)] = x ∗ [x ∗ θ(x)] for all x, y ∈ X.

Theorem 5.2 Let θ be an anti-self preserving left monic on an associative obic algebra X.

Then 0 ∗ [θ(x) ∗ x] = θ(0). Moreover, if θ is regular, then 0 ∗ [θ(x) ∗ x] = 0 for all x ∈ X.

Proof Notice that

θ(0) = θ(x ∗ x) = [θ(x) ∗ x] ∧ [x ∗ θ(x)]= [θ(x) ∗ x] ∧ x= 0 ∗ [θ(x) ∗ x].

The second part of the theorem is obvious. 2

88 Ilojide Emmanuel

Proposition 5.10 Let X be an obic algebra with a left monic θ. Then θ(x) = x ∗ [x ∗ θ(x)] for

all x ∈ X.

Proof Notice that

θ(x) = θ(x ∗ 0) = θ(x)

= [θ(x) ∗ 0] ∧ [x ∗ θ(0)]

= x ∗ [x ∗ θ(x)].

This completes the proof. 2Corollary 5.2 Let θ be an anti-self preserving regular left monic on an obic algebra X. Then

θ(x) = 0 for all x ∈ X.

Corollary 5.3 Let θ be a regular left monic on an associative obic algebra X. Then θ(x) =

0 ∗ θ(x) for all x ∈ X.

References

[1] H. S. Kim and Y. H. Kim, On BE-algebras, Sci. Math. Jpn., 66(2007),113–116.

[2] S.S. Ahn and K. S. So, On ideals and upper sets in BE-algebras, Sci. Math. Jpn., 68(2008),

351–357.

[3] S.S. Ahn and K. S. So, On generalized upper sets in BE-algebras, Bull. Korean Math.

Soc., 46(2009), 281–287.

[4] R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, Berlin-Gottingen-Heidelberg,

1966, 185pp.

[5] J. Dene and A. D. Keedwell, Latin Squares and Their Applications, the English University

press Ltd, 1974, 549pp.


Prime BI-Ideals of Po-Γ-Groupoids

J.Catherine Grace John and B. Elavarasan

(Department of Mathematics, Karunya University, Coimbatore-641114, India)


Abstract: In this paper, we have studied the notion of prime bi-ideals and semi prime

bi-ideals of Po-Γ-groupoids and explored the various properties of prime bi-ideals in Po-Γ-

groupoids. Also we obtained the condition for Po-Γ-groupoids to be regular.

Key Words: Partially ordered Γ-groupoid (po-Γ-groupoid), left (right, two-sided) ideal,

quasi-ideal, bi-ideal, prime (semi prime) bi-ideal, regular partially ordered -Γ-groupoid.

AMS(2010): 06F05, 20N02.

§1. Introduction

In 1983, A.P.J. van der Walt [5] introduced the interesting concepts of prime and semi prime

bi-ideals for an associative ring with unity. In 1995, using the concepts defined by A. P. J.

van der Walt, the structure of a ring containing prime and semi prime bi-ideals were studied

by H. J. le Roux [2]. In 2001, Kehayopulu and Tsingelis [1] studied prime ideals of groupoids.

Following [1], in 2005, S.K. Lee developed prime left (right) ideals of groupoids [3] and obtained

some results on prime bi-ideals of groupoids [4]. In this paper we have studied the notion of

prime bi-ideals and semi prime bi-ideals of Po-Γ-groupoids. Let M be a non empty set. M is

called Γ-groupoid if for all a, b ∈M and γ ∈ Γ, aγb ∈M .

A set (G,Γ,≤) is called a partial order-Γ-groupoid(or simply Po-Γ-groupoid) if

(i) (G,≤) is a partial ordered set;

(ii) (G,Γ) is a Γ-groupoid such that a ≤ b ⇒ aγx ≤ bγx and xγ1a ≤ xγ1b for all a, b, x ∈G; γ, γ1 ∈ Γ.

Throughout this paper G denotes a Po-Γ-groupiod.

A non empty subset A of G is called right(resp.left) ideal of G if

(i) AΓG ⊆ A ( resp.GΓA ⊆ A);

(ii) a ∈ A , b ≤ a for b ∈ G implies b ∈ A.

A non empty subset A is called an ideal of G if it is a right and left ideal of G.

For non-empty subsets A and B of a po-Γ-groupoid G, the product AΓB of A and B

and the subset (A] of G are defined by AΓB = {aγb ∈ S : a ∈ A, b ∈ B, γ ∈ Γ}; (A]=

{x ∈ G : ∃a ∈ A(x ≤ a)}.1Received July 31, 2019, Accepted December 3, 2019.

90 J.Catherine Grace John and B. Elavarasan

A non empty subset Q of G is called a quasi ideal if

(i) (QΓG] ∩ (GΓQ] ⊆ Q;

(ii) a ≤ q; q ∈ Q implies a ∈ Q.

A non empty subset B of G is called a bi-ideal if

(i) (BΓGΓB] ⊆ B;

(ii) a ≤ b; b ∈ B implies a ∈ B.

Every quasi ideal is a bi-ideal. But the converse need not be true. A bi-ideal B of G is prime,

for x, y ∈ G, (xΓGΓy] ⊆ B implies x ∈ B or y ∈ B. A bi-ideal B of G is semi-prime, for

x ∈ G, (xΓGΓx] ⊆ B implies x ∈ B. A non-empty subset I of G is prime if I is an ideal

of G such that for any ideals A,B of G , AB ⊆ I implies A ⊆ I or B ⊆ I. It is clear that

(x)l = (x ∪GΓx](resp.(x)r = (x ∪ xΓG]) is the principle left(resp.right) ideal generated by x.

§2. Main Results

Theorem 2.1 A bi-ideal B of G is prime if and only if for a right ideal R and a left ideal L

of G (RΓL] ⊆ B implies R ⊆ B or L ⊆ B.

Proof Suppose that (RΓL] ⊆ B for a right ideal R and a left ideal L of G and R * B.

Then there exists x ∈ R\B such that (xΓGΓy] ⊆ (RΓGΓL] ⊆ (RΓL] ⊆ B for any y ∈ L which

implies y ∈ B. So L ⊆ B.

Conversely, let (xΓGΓy] ⊆ B for x, y ∈ G. Then (xΓG]Γ(GΓy] ⊆ (xΓGΓGΓy] ⊆ (xΓGΓy] ⊆B. By hypothesis, we have (xΓG] ⊆ B or GΓy] ⊆ B. If (xΓG] ⊆ B, then xΓx ∈ (xΓG]ΓG ⊆ B.

Now, (x)r(x)l = (x∪xΓG]Γ(x∪GΓx] = (xΓx∪xΓGΓx∪xΓGΓx∪xΓGΓGΓx] ⊆ (xΓx∪xΓG] ⊆ B

which implies (x)r ⊆ B or (x)l ⊆ B. Therefore x ∈ B. If (GΓy] ⊆ B, then by the similar

method y ∈ B. 2Theorem 2.2 If a bi-ideal B of G is prime, then B is a left or right ideal of G.

Proof Let B be a prime bi-ideal of G.Then (BΓG] ⊆ B or (GΓB] ⊆ B as (BΓG]Γ(GΓB] ⊆(BΓGΓB] ⊆ B and by Theorem 2.1. So B is a a left ideal or right ideal of G. 2Theorem 2.3 Let G be a po-Γ-groupoid. Then the following statements are hold:

(i) Any left/right/both sided ideal of G is a bi-ideal of G;

(ii) Intersection of right and left ideals of G is a bi-ideal of G;

(iii) Arbitrary intersection of bi-ideals of G is also a bi-ideal of G;

(iv) If B is a bi-ideal of G, then BΓr and rΓB are bi-ideals of G, for any r ∈ G.

Proof This result can be immediately verified by definition. 2Notation 1 For a bi-ideal of B of G, we define LB = {x ∈ B : GΓx ⊆ B}, RB = {x ∈B : xΓG ⊆ B}, IL = {y ∈ LB : yΓG ⊆ LB} and IR = {y ∈ RB : GΓy ⊆ RB}.

Prime BI-Ideals of Po-Γ-Groupoids 91

Theorem 2.4 Let B be bi-ideal of G . Then LB is a left ideal of G contained in B if LB is

non empty.

Proof Let x ∈ LB. Let g ∈ G and γ ∈ Γ . Then gγx ∈ GΓx ⊆ B . Now GΓgγx ⊆GΓGΓx ⊆ GΓx ⊆ B which implies gγx ∈ LB. GΓLB ⊆ LB. hence LB is a left ideal. 2Theorem 2.5 Let B be bi-ideal of G. Then IL is the largest ideal of G contained in B if IL is

non empty. Furthermore, IL coincides with IR.

Proof Let x ∈ IL.Then xΓG ⊆ LB. For any g ∈ G and γ ∈ Γ, we have xγg ∈ xΓG ⊆ LB

and xγgΓG ⊆ xΓGΓG ⊆ xΓG ⊆ LB, So IL is a right ideal of G.

Since IL ⊆ LB ⊆ B, we have x ∈ LB which implies xγg ∈ IL and GΓx ⊆ B.

Now, GΓgγx ⊆ GΓGΓx ⊆ GΓx ⊆ B. So gγx ∈ LB. By Theorem 2.4 and x ∈ IL, we have

xΓG ⊆ LB. Then gγxΓG ⊆ GΓLB ⊆ LB, and we have gγx ∈ IL. Therefore IL is a left ideal.

Let A be an ideal of G such that A ⊆ B. If x ∈ A, then x ∈ B and GΓ ⊆ x ⊆ A ⊆ B

which implies x ∈ LB and A ⊆ LB.

Let x ∈ A. Then xΓG ⊆ A ⊆ LB. Hence x ∈ IL and A ⊆ IL which implies IL is the

largest ideal of G contained in B. Similarly IR is the largest ideal of G contained in B. 2Notation 2 We denote IB as IB :≡ IR = IL by Theorem 2.5.

Theorem 2.6 If B is a prime bi-ideal of G, then IB is a prime ideal of G contained in B.

Proof Let B be a prime bi-ideal of G. Then by Theorem 2.5, IB is an ideal of G.

Suppose XΓY ⊆ IB for any ideals X,Y of G. Since IB ⊆ LB ⊆ B, we have XΓY ⊆ B.

By Theorem 2.1, X ⊆ B or Y ⊆ B. But IB is the largest ideal contained in B, so X ⊆ IB or

Y ⊆ IB which implies IB is a prime ideal of G. 2Corollary 2.7 If B be a semi-prime bi-ideal of G, then IB is a semi-prime ideal of G if IB is

non empty.

Theorem 2.8 If a bi-ideals B of G is semi-prime, then

(i) for any left ideal L of G, LΓL ⊆ B implies L ⊆ B;

(ii) for any right ideal R of G, RΓR ⊆ B implies R ⊆ B.

Proof Suppose LΓL ⊆ B for a left ideal L of G and L * B. Then there exists x ∈ L\B,xΓGΓx ⊆ LΓGΓL ⊆ LΓL ⊆ B. Since B is a semi-prime we have x ∈ B, a contradiction.

The second assertion can be proved similarly. 2Theorem 2.9 If a bi-ideal B of G is semi-prime, then B is a quasi-ideal of G.

Proof Let y ∈ (BΓG] ∩ (GΓB]. Then (yΓGΓy] ⊆ ((BΓG]ΓGΓ(GΓB]] ⊆ (BΓGΓB] ⊆ B.

Since B is a semi prime, we have y ∈ B. Hence B is a quasi-ideal of G. 2Remark 2.10 For a Po-Γ-groupoid G,

(a) The set of all prime ideal of G is denoted by spec(G);

92 J.Catherine Grace John and B. Elavarasan

(b) Bspec(G) denotes the set of prime bi-ideals of G;

(c) Sspec(G) denotes the set of all semi prime bi-ideals of G.

Then we know conclusions following easily by definition.

Theorem 2.11 If G is finite, then⋂spec(G) =

⋂Bspec(G).

Theorem 2.12 A bi-ideal B of G is semi prime if and only if for a right ideal ( left ideal) A

of G (AΓA] ⊆ B implies A ⊆ B.

Theorem 2.13 The intersection of any family of prime bi-ideals of G is a semi prime bi-ideal

of G.

Theorem 2.14 If G is finite, then⋂spec(G) =

⋂Sspec(G).

We note that G is regular if for any x ∈ G, there exist a ∈ G and γ1, γ2 ∈ Γ such that

x ≤ xγ1aγ2x.

The following results shows the necessary and sufficient condition for a Po-Γ-groupiod to

be regular.

Theorem 2.15 Let G be Po-gamma-groupiod. Then G is regular if and only if every bi-ideal

of G is semi-prime.

Proof Let G be regular and B a bi-ideal of G. Suppose that xΓGΓx ⊆ B for x ∈ G. Then

there exist a ∈ G and γ1, γ2 ∈ Γ such that x ≤ xγ1aγ2x ∈ xΓaΓx ∈ xΓGΓx ⊆ B which implies

x ∈ B. Hence B is semi prime.

Conversely , assume that every bi-ideal of G is semi-prime. Let B = (aΓGΓa] for a ∈ G.

Then BΓGΓB = (aΓGΓa]ΓGΓ(aΓGΓa] ⊆ (aΓGΓa] = B, which implies B is a bi-ideal of G and

by assumption (aΓGΓa] is semi-prime. Since aΓGΓa ⊆ (aΓGΓa] = B, we get a ∈ (aΓGΓa] = B.

Then there exist x ∈ G and γ1, γ2 ∈ Γ such that a ≤ aγ1xγ2a. 2References

[1] N. Kehayopulu, On prime ideals of groupoids-ordered groupoids, Scientiae Mathematicae

Japonicae, 5 (2001), 83-86.

[2] H. J. le Roux, A note on prime and semiprime bi-ideals of rings, Kyungpook Math. J., 35

(1995), 243-247.

[3] S.K.Lee, On prime left (right) ideals of grou[iogs- ordered groupiods, Kangweon-Kyungki

Math. J., 13(1) (2005), 13-18.

[4] S.K.Lee, Prime bi-ideals groupiods, Kangweon- Kyungki Math. J., 13(2) (2005), 217-221.

[5] A. P. J. van der Walt, Prime and semiprime bi-ideals,Quaestiones Matematicae, 5 (1983),

341-345.


4-Total Prime Cordiality of Certain Subdivided Graphs

R. Ponraj

(Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-627412, India)

J. Maruthamani and R. Kala

(Department of Mathematics, Manonmaniam Sundarnar University, Tirunelveli-627012, Tamilnadu, India)

E-mail: [email protected], [email protected],[email protected]

Abstract: Let G be a (p, q) graph. Let f : V (G) → {1, 2, · · · , k} be a map where k ∈ N

and k > 1. For each edge uv, assign the label gcd(f(u), f(v)). f is called k-total prime

cordial labeling of G if |tf (i) − tf (j)| ≤ 1, i, j ∈ {1, 2, · · · , k} where tf (x) denotes the total

number of vertices and the edges labelled with x. A graph with a k-total prime cordial

labeling is called k-total prime cordial graph. In this paper we investigate the 4-total prime

cordial labeling for some subdivided graphs.

Key Words: Corona, ladder, triangular snake, k-total prime cordial labeling, Smarandache

k-total prime cordial labeling.

AMS(2010): 05C78.

§1. Introduction

In this paper we consider simple, finite and undirected graphs only. The notion of k- total

prime cordial labelling has been introduced in [4]. In [4–9], they investigate the k-total prime

cordial labeling of some graphs and investigate the 4-total prime cordial labeling behaviour of

path, cycle, star, bistar, ladder, triangular snake, friendship graph, comb, double comb, double

triangular snake, flower graph, gear graph, Jelly fish, book, irregular triangular snake, prism,

helm, dumbbell graph, sunflower graph, dragon, mobius ladder and subdivision of some graphs.

In this paper we examine the 4-total prime cordial labeling of subdivision of some graphs like

star, bistar, comb, double comb, ladder, triangular snake and double triangular snake. Terms

are not defined here follows from [1], [3].

§2. Preliminary Results

Definition 2.1 Let G1, G2 respectively be (p1, q1), (p2, q2) graphs. The corona of G1 with G2,

G1 ⊙G2 is the graph obtained by taking one copy of G1 and p1 copies of G2 and joining the ith

vertex of G1 with an edge to every vertex in the ith copy of G2.

1Received April 14, 2019, Accepted December 5, 2019.

94 R. Ponraj, J. Maruthamani and R. Kala

Definition 2.2 If e = uv is an edge of G then e is said to be subdivided when it is replaced by

the edges uw and wv. The graph obtained by subdividing each edge of a graph G is called the

subdivision graph of G and is denoted by S(G).

§3. k-Total Prime Cordial Labeling

Definition 3.1 Let G be a (p, q) graph. Let f : V (G) → {1, 2, · · · , k} be a function where

k ∈ N and k > 1. For each edge uv, assign the label gcd(f(u), f(v)). f is called k-total

prime cordial labeling of G if |tf (i) − tf (j)| ≤ 1, i, j ∈ {1, 2, · · · , k} where tf (x) denotes the

total number of vertices and the edges labelled with x. A graph with a k-total prime cordial

labeling is called k-total prime cordial graph. Generally, if there are integers i, j ∈ {1, 2, · · · , k}such that |tf (i) − tf (j)| > 1, f is called a Smarandache k-total prime cordial labeling and G a

Smarandache k-total prime cordial labeling graph.

Theorem 3.2 The subdivision of comb S(Pn ⊙K1) is 4-total prime cordial.

Proof Let Pn be the path u1u2 · · ·un. Let xi be the vertex which subdivide the edge

uiui+1. Let vi be the vertex adjacent to ui. Let wi be the pendent vertices vi. Clearly

|V (S(Pn ⊙K1))| + |E(S(Pn ⊙K1))| = 8n− 3.

Case 1. n ≡ 0 (mod 4).

Let n = 4t, t ∈ N. Assign the label 4 to the vertices u1, u2, · · · , ut and assign the label 3 to

the vertices ut+1, ut+2, · · · , u2t. Next we assign the label 2 to the vertices u2t+1, u2t+2, · · · , u3t

and assign 1 to the vertices u3t+1, u3t+2, · · · , un. Now we consider the vertices xi (1 ≤ i ≤n − 1). Assign the label 4 to the vertices x1, x2, · · · , xt and assign the label 3 to the vertices

xt+1, xt+2, . . . , x2t. Next we assign the label 2 to the vertices x2t+1, x2t+2, · · · , x3t. Finally we

assign 1 to the vertices x3t+1, x3t+2, · · · , xn−1. Now we move to the vertices vi, wi (1 ≤ i ≤ n).

Assign the label 4 to the vertices v1, v2, . . . , vt and w1, w2, · · · , wt. Then assign the label 3

to the vertices vt+1, vt+2, · · · , v2t and wt+1, wt+2, · · · , w2t. Now we assign the label 2 to the

vertices v2t+1, v2t+2, · · · , v3t and w2t+1, w2t+1, · · · , w3t. Finally we assign the label 1 to the

vertices v3t+1, v3t+2, · · · , vn and w3t+1, w3t+2, · · · , wn.

Case 2. n ≡ 1 (mod 4).

Let n = 4t+ 1, t ∈ N. As in case 1, assign the label to the vertices ui (1 ≤ i ≤ n− 2), xi

(1 ≤ i ≤ n − 2), vi, wi (1 ≤ i ≤ n − 1). Now we assign the labels 3, 4, 2, 3, 4 respectively to

the vertices un−1, un, xn−1, vn and wn.

Case 3. n ≡ 2 (mod 4).

Let n = 4t + 2, t ∈ N. Assign the label to the vertices ui, vi, wi (1 ≤ i ≤ n − 2) and xi

(1 ≤ i ≤ n− 3) by case 1. Next we assign the labels 4, 3, 4, 3, 2, 2, 4, 3 to the vertices un−1,

un, xn−2, xn−1, vn−1, vn, wn−1 and wn respectively.

Case 4. n ≡ 3 (mod 4).

4-Total Prime Cordiality of Certain Subdivided Graphs 95

Let n = 4t+3, t ∈ N. As in Case 3, assign the label to the vertices ui, vi, wi (1 ≤ i ≤ n−2)

and xi (1 ≤ i ≤ n− 3). Next we assign the labels 2, 3, 4, 3, 3, 3, 2,1 respectively to the vertices

un−1, un, xn−2, xn−1, vn−1, vn, wn−1 and wn. 2Theorem 3.3 The subdivision of double comb S(Pn ⊙ 2K1) is 4-total prime cordial.

Proof Let Pn be the path u1u2 · · ·un. Let zi be the vertex which subdivide the edge

uiui+1. Let vi, wi be the vertices adjacent to ui. Let xi, yi be the pendent vertices adjacent to

ui,vi respectively. Obviously |V (S(Pn ⊙ 2K1))| + |E(S(Pn ⊙ 2K1))| = 12n− 3.

Case 1. n ≡ 0 (mod 4).


the vertices ut+1, ut+2, · · · , u2t. Now we assign the label 2 to the vertices u2t+1, u2t+2, · · · , u3t.

Next we assign 1 to the vertices u3t+1, u3t+2, · · · , un−1. Finally we assign the label 3 to the

vertex un. Now we consider the vertices zi (1 ≤ i ≤ n − 1). Assign the label 4 to the vertices

z1, z2, · · · , zt−1 and assign the label 3 to the vertices zt+1, zt+2, · · · , z2t−1. Next we assign

the label 2 to the vertices z2t+1, z2t+2, · · · , z3t−1. Now we assign the label 1 to the vertices

zt, z2t, z3t, z3t+1, · · · , zn−1. Assign the label to the vertices vi, wi, xi, yi (1 ≤ i ≤ n−1). Finally

we assign 2, 4, 4, 3 respectively to the vertices xn, vn, wn and yn.

Case 2. n ≡ 1 (mod 4).

Let n = 4t + 1, t ∈ N. As in case 1, assign the label to the vertices ui, vi, xi, wi, yi

(1 ≤ i ≤ n− 1) and zi (1 ≤ i ≤ n− 2). Now we assign the labels 2, 4, 4, 2, 3, 3 respectively to

the vertices zn−1, xn, vn, un, wn and yn.

Case 3. n ≡ 2 (mod 4).

Let n = 4t+ 2, t ∈ N. Assign the label to the vertices ui, vi, xi, wi, yi (1 ≤ i ≤ n− 1) and

zi (1 ≤ i ≤ n − 2) by case 2. Next we assign the label 1 to zn−1 and assign the labels 4, 4, 2,

3, 3 to the vertices xn, vn, un, wn and yn respectively.

Case 4. n ≡ 3 (mod 4).

Let n = 4t + 3, t ∈ N. As in Case 3, assign the label to the vertices ui, vi, xi, wi, yi

(1 ≤ i ≤ n− 1) and zi (1 ≤ i ≤ n− 2). Next we assign the labels 2, 4, 4, 2, 3, 3 respectively to

the vertices zn−1, xn, vn, un, wn and yn. 2Theorem 3.4 The subdivision of star S(K1,n) is 4-total prime cordial.

Proof Let u be the vertex of degree n and u1, u2, · · · , un be the vertices of degree 2. Let

v1, v2, · · · , vn be the pendent vertices.

Case 1. n ≡ 0 (mod 4).

Let n = 4t, t ∈ N. Assign the label 4 to the vertex u. Next we now move to the vertices

u1, u2, · · · , un. Assign the label 4 to the vertices u1, u2, · · · , ut and assign the label 2 to the

vertices ut+1, ut+2, · · · , u2t. Next we assign the label 3 to the vertices u2t+1, u2t+2, · · · , u3t.

Finally assign the label 1 to the non-labelled vertices of un. Now we consider the pendent


vertices v1, v2, · · · , vn. Assign the label 4 to the vertices v1, v2, · · · , vt and assign the label 2 to

the vertices vt+1, vt+2, . . . , v2t. Finally assign 3 to the non-labelled vertices of vn.

Case 2. n ≡ 1 (mod 4).

Let n = 4t + 1, t ∈ N. In this case, assign the label to the vertices u, ui (1 ≤ i ≤ n) and

vi (1 ≤ i ≤ n − 2) by in case 1. Next assign the labels 2 and 4 to the vertices vn−1 and vn

respectively.

Case 3. n ≡ 2 (mod 4).

Let n = 4t+ 2, t ∈ N. As in Case 1, assign the label to the vertices u, ui (1 ≤ i ≤ n− 1)

and vi (1 ≤ i ≤ n− 2). Next assign the labels 2, 1 and 4 to the vertices respectively un, vn−1

and vn.

Case 4. n ≡ 3 (mod 4).

Let n = 4t + 3, t ∈ N. Assign the label to the vertices u, ui (1 ≤ i ≤ n − 2) and vi

(1 ≤ i ≤ n − 2) as in case 1. Finally assign the labels 3, 2, 4 and 4 to the vertices un−1, un,

vn−1 and vn respectively. 2Theorem 3.5 The subdivision of bistar S(Bn,n) is 4-total prime cordial.

Proof Let u, v be the vertices of degree n and w be the vertex of degree 2 adjacent to both

u and v. Let ui be the vertex of degree 2 adjacent to u and vi be the vertex of degree 2 adjacent

to v. Let xi and yi (1 ≤ i ≤ n) be the pendent vertex adjacent to ui and vi respectively.

Case 1. n ≡ 0 (mod 4).

Let n = 4t, t ∈ N. Assign the labels 4, 2 and 3 to the vertex u, w and v respectively. Next

we move to the vertices u1, u2, · · · , un. Assign the label 4 to the vertices u1, u2, · · · , u2t and

assign the label 2 to the vertices u2t+1, u2t+2, · · · , u4t. Now we consider the pendant vertices

of un. Assign the label 4 to the vertices x1, x2, · · · , x2t and assign the label 2 to the vertices

x2t+1, x2t+2, · · · , x4t. Now we consider the vertices v1, v2, · · · , vn. Assign the label 3 to the

vertices v1, v2, · · · , v2t and assign the label 1 to the vertices v2t+1, v2t+2, · · · , v4t. Finally we

move to the pendant vertices of vn. Assign the label 3 to the vertices y1, y2, · · · , y2t and assign

the label 1 to the vertices y2t+1, y2t+2, · · · , y4t.

Case 2. n ≡ 1 (mod 4).

Let n = 4t+1, t ∈ N. In this case assign the label to the vertices u, v, w, ui (1 ≤ i ≤ n−1),

vi (1 ≤ i ≤ n− 1), xi (1 ≤ i ≤ n− 1) and yi (1 ≤ i ≤ n− 1) as in case 1. Next assign the labels

4, 2, 3 and 1 respectively to the vertices un, xn, vn and yn.

Case 3. n ≡ 2, 3 (mod 4).

Let n = 4t+ 1 and n = 4t+ 2 t ∈ N. The proof is similar to that of Case 2. 2Theorem 3.6 The subdivision of triangular snake S(Tn) is 4-total prime cordial.

Proof Let Pn be the path u1u2 · · ·un. Let wi be the vertex adjacent to ui and ui+1. Let


vi be the vertices which subdivide the edge uiui+1 and xi,yi be the vertex which subdivided

uiwi and ui+1wi respectively. It is easy to verify that |V (S(Tn))| + |E(S(Tn))| = 11n− 10.

Case 1. n ≡ 0 (mod 4).



then we assign the label 1 to the vertices u3t+1, u3t+2, · · · , un−1. Finally, we assign 3 to the

vertex un. Assign the label 4 to the vertices v1, v2, · · · , vt and assign the label 2 to the vertices

vt+1, vt+2, · · · , v2t−1 and assign the label 3 to the vertices v2t, v2t+1, · · · , v3t−1 then we assign

the label 1 to the vertices v3t, v3t+1, · · · , vn−1. Assign the label to the vertices xi (1 ≤ i ≤ n−1)

as in vi (1 ≤ i ≤ n − 1). Now relabel the vertex x2t by 2. Assign the label 4 to the vertices

y1, y2, · · · , yt−1 and assign the label 2 to the vertices yt, yt+1, · · · , y2t−1 and assign the label 3 to

the vertices y2t, y2t+1, · · · , y3t−1 then we assign the label 1 to the vertices y3t, y3t+1, · · · , yn−2.

Finally we assign the label 2 to the vertices yn−1. Now we consider the vertices wi (1 ≤ i ≤n− 1). Assign the label 4 to the vertices w1, w2, · · · , wt and assign the label 2 to the vertices

wt+1, wt+2, · · · , w2t−1 and assign the label 3 to the vertices w2t, w2t+1, · · · , w3t−1 then we assign

the label 1 to the vertices w3t, w3t+1, · · · , wn−2. Finally we assign 4 to the vertex wn−1.

Case 2. n ≡ 1 (mod 4).

Let n = 4t+ 1, t ∈ N. As in Case 1, assign the label to the vertices ui (1 ≤ i ≤ n− 1), vi,

xi, yi, wi (1 ≤ i ≤ n− 2). Next we assign the labels 3, 3, 2, 4, 4 to the vertices vn−1, un, xn−1,

yn−1 and wn−1 respectively.

Case 3. n ≡ 2 (mod 4).

Let n = 4t + 2, t ∈ N. Assign the label to the vertices ui (1 ≤ i ≤ n − 3), vi, xi, wi

(1 ≤ i ≤ n−3) and yi (1 ≤ i ≤ n−4) as in Case 2. Now we assign the labels 4, 3, 3 respectively

to the vertices un−2, un−1 and un. Next we assign the labels to the vertices 4, 3, 2, 2, 2, 1 to

the vertices vn−2, vn−1, xn−2, xn−1, wn−2, and wn−1 respectively. Finally we assign the labels

4, 2, 3 respectively to the vertices yn−3, yn−2 and yn−1.

Case 4. n ≡ 3 (mod 4).

Let n = 4t + 3, t ∈ N. Assign the label to the vertices ui, xi, yi, wi (1 ≤ i ≤ n − 3) and

vi (1 ≤ i ≤ n − 4) as in Case 3. Now we assign the labels 4, 3, 3, 3, 1, 1, 3, 4, 3 respectively

to the vertices un−2, un−1, un, xn−2, xn−1, yn−2, yn−1, wn−2 and wn−2. Finally we assign the

labels 2, 4, 3 to the vertices vn−3, vn−2 and vn−1 respectively. 2Theorem 3.7 The subdivision of ladder S(Ln) is 4-total prime cordial.

Proof Let V (Ln) = {ui, vi : 1 ≤ i ≤ n} and E(Ln) = {uiui+1, vivi+1 : 1 ≤ i ≤ n − 1} ∪{uivi : 1 ≤ i ≤ n}. Let yi, wi and xi be the vertices which subdivide the edges uiui+1, uivi and

vivi+1 respectively. Clearly |V (S(Ln))| + |E(S(Ln))| = 11n− 6.

Case 1. n ≡ 0 (mod 4).

Let n = 4t, t ∈ N. Assign the label 4 to the vertices u1, u2, · · · , ut and v1, v2, · · · , vt.


Assign the label 2 to the vertices ut+1, ut+2, · · · , u2t and vt+1, vt+2, · · · , v2t. Next we assign

the label 3 to the vertices u2t+1, u2t+2,. . . , u3t and v2t+1, v2t+2, · · · , v3t then we assign the

label 1 to the vertices u3t+1, u3t+2, · · · , un−1 and v3t+1, v3t+2, · · · , vn−1. Finally, we assign

the labels 4 and 3 to the vertices un and vn respectively. Next we consider the vertices xi

(1 ≤ i ≤ n). Assign the label 4 to the vertices x1, x2, · · · , xt and assign the label 2 to the

vertices xt+1, xt+2, · · · , x2t. Now we assign the label 3 to the vertices x2t+1, x2t+2, · · · , x3t.

Finally we assign the label 1 to the vertices x3t+1, x3t+2, · · · , xn. Now we consider the vertices

yi, wi (1 ≤ i ≤ n−1). Assign the label 4 to the vertices y1, y2, · · · , yt and w1, w2, · · · , wt. Assign

the label 2 to the vertices yt+1, yt+2, · · · , y2t−1 and wt+1, wt+2, · · · , w2t−1. Next we assign the

label 3 to the vertices y2t, y2t+1, · · · , y3t−1 and w2t, w2t+1, · · · , w3t−1 then we assign the label

1 to the vertices y3t, y3t+1, · · · , yn−2 and w3t, w3t+1,· · · , wn−2, wn−1. Finally, we assign the

labels 2 vertex yn−1.

Case 2. n ≡ 1 (mod 4).

Let n = 4t+ 1, t ∈ N. As in Case 1, assign the label to the vertices ui (1 ≤ i ≤ n− 1), vi

(1 ≤ i ≤ n − 1), xi (1 ≤ i ≤ n), yi (1 ≤ i ≤ n − 2) and wi (1 ≤ i ≤ n − 2). Finally we assign

the labels 2, 2, 4, 3 respectively to the vertices un, vn, yn−1 and wn−1.

Case 3. n ≡ 2 (mod 4).


(1 ≤ i ≤ n− 1), xi (1 ≤ i ≤ n− 1), yi (1 ≤ i ≤ n− 2) and wi (1 ≤ i ≤ n− 2). Finally we assign

the labels 4, 3, 2, 4, 3 to the vertices un, vn, xn, yn−1 and wn−1 respectively.

Case 4. n ≡ 3 (mod 4).


(1 ≤ i ≤ n− 1), xi (1 ≤ i ≤ n− 1), yi (1 ≤ i ≤ n− 2) and wi (1 ≤ i ≤ n− 2). Finally we assign

the labels 3, 4, 2, 3, 4 respectively to the vertices un, vn, xn, yn−1 and wn−1. 2Theorem 3.8 The subdivision of double triangular snake S(DTn) is 4-total prime cordial.

Proof Let Pn be the path u1u2 · · ·un. Let vi, wi be the vertex adjacent to uiui+1. Let xi,

yi, zi, si and ri be the vertex which subdivide the edges uiui+1, uivi, viui+1, uiwi and wiui+1

respectively. Clearly |V (S(DTn))| + |E(S(DTn))| = 18n− 17.

Case 1. n ≡ 0 (mod 4), n ≥ 8.



then we assign the label 1 to the vertices u3t+1, u3t+2, · · · , un−1. Finally, we assign the label 3

to the vertex un. Now we consider the vertices vi,wi (1 ≤ i ≤ n− 1). Assign the label 4 to the

vertices v1, v2, · · · , vt and w1, w2, · · · , wt. Assign the label 2 to the vertices vt+1, vt+2, · · · , v2t−1

and wt+1, wt+2, · · · , w2t−1. Next we assign the label 3 to the vertices v2t, v2t+1, · · · , v3t−1

and w2t, w2t+1, · · · , w3t−1 then we assign the label 1 to the vertices v3t, y3t+1, · · · , vn−3 and

w3t, w3t+1, · · · , wn−3. Finally we assign the labels 2, 4, 2, 4 respectively to the vertices vn−2,

vn−1, wn−2 and wn−1. Next we move to the vertices xi (1 ≤ i ≤ n− 1). Assign the label 4 to


the vertices x1, x2, · · · , xt and assign the label 2 to the vertices xt+1, ut+2, · · · , x2t−1. Next we

assign the label 3 to the vertices x2t, x2t+1, · · · , x3t−1 then we assign the label 1 to the vertices

x3t, x3t+1, · · · , xn−2. Finally, we assign the label 3 to the vertex xn−1. Now we consider the

vertices yi,si (1 ≤ i ≤ n− 1). Assign the label 4 to the vertices y1, y2, · · · , yt and s1, s2, · · · , st.

Assign the label 2 to the vertices yt+1, yt+2, · · · , y2t and st+1, st+2, · · · , s2t. Next we assign the

label 3 to the vertices y2t+1, y2t+2, · · · , y3t−1 and s2t+1, s2t+2, · · · , s3t−1. Finally we assign the

label 1 to the vertices y3t, y3t+1, · · · , yn−1 and s3t, s3t+1, · · · , sn−1. Next we move to the vertices

zi, ri (1 ≤ i ≤ n − 1). Assign the label 4 to the vertices z1, z2, · · · , zt−1 and r1, r2, · · · , rt−1.

Assign the label 2 to the vertices zt, zt+1, · · · , z2t−1 and rt, rt+1, · · · , r2t−1. Next we assign the

label 3 to the vertices z2t, z2t+1, · · · , z3t−1 and r2t, r2t+1, · · · , r3t−1. Finally we assign the label

1 to the vertices z3t, z3t+1, · · · , zn−1 and r3t, r3t+1, · · · , rn−1. Clearly tf (1) = tf (2) = tf (3) =

18t− 4 and tf (4) = 18t− 5.

Case 2. n ≡ 1 (mod 4), n ≥ 9.

Let n = 4t+ 1, t ∈ N. As in Case 1, assign the label to the vertices ui (1 ≤ i ≤ n− 1), vi,

wi, xi, yi, zi, si, ri (1 ≤ i ≤ n− 2). Finally we assign the labels 4, 4, 2, 3, 1, 4, 3, 2 respectively

to the vertices un, vn−1, wn−1, xn−1, yn−1, zn−1, sn−1 and rn−1. Obviously tf (1) = 18t + 1

and tf (2) = tf (3) = tf (4) = 18t.

Case 3. n ≡ 2 (mod 4), n ≥ 10.

Let n = 4t+ 2, t ∈ N. Assign the label to the vertices ui (1 ≤ i ≤ n− 2), vi, wi, xi, yi, zi

(1 ≤ i ≤ n− 3), si, ri (1 ≤ i ≤ n− 2) by in case 1. Finally we assign the labels 2, 4, 3, 4, 3, 3,

1, 2, 3, 4, 2, 4, 1, 4 to the vertices un−1, un, vn−2, vn−1, wn−2, wn−1, xn−2, xn−1, yn−2, yn−1,

zn−2, zn−1, sn−1 and rn−1 respectively. It is easy to verify that tf (1) = tf (2) = tf (3) = 18t+5

and tf (4) = 18t+ 4.

Case 4. n ≡ 3 (mod 4), n ≥ 11.

Let n = 4t + 3, t ∈ N. As in Case 3, assign the label to the vertices ui (1 ≤ i ≤ n − 1),

vi, wi, xi, yi, zi, si, ri (1 ≤ i ≤ n − 2). Finally we assign the labels 3, 3, 2, 2, 4, 3, 4,

1 respectively to the vertices un, vn−1, wn−1, xn−1, yn−1, zn−1, sn−1 and rn−1. Clearly

tf (1) = tf (2) = tf (4) = 18t+ 9 and tf (3) = 18t+ 10.

Case 5. t = 2, 3, 4, 5, 6, 7.

A 4-total prime cordial labeling is given in Table 1.

n 2 3 4 5 6 7

u1 3 4 4 4 4 4

u2 4 2 2 4 4 4

u3 3 3 2 2 2

u4 3 3 3 2

u5 1 1 3


u6 2 1

u7 1

v1 4 4 4 4 4 4

v2 3 3 2 2 4

v3 4 3 3 2

v4 3 3 3

v5 4 3

v6 1

w1 2 4 4 4 4 4

w2 1 2 2 2 4

w3 1 3 3 2

w4 1 1 3

w5 3 3

w6 1

x1 1 2 4 4 4 4

x2 3 2 2 2 2

x3 3 3 3 2

x4 3 3 3

x5 4 3

x6 1

y1 3 4 4 4 4 4

y2 3 3 2 2 4

y3 1 3 3 2

y4 1 1 3

y5 3 3

y6 1

z1 4 2 2 4 4 4

z2 3 3 2 2 2

z3 4 3 3 2

z4 1 1 3

z5 4 1

z6 1

s1 3 4 4 4 4 4

s2 1 2 2 2 4


s3 1 3 3 2

s4 1 1 3

s5 3 3

s6 1

r1 2 2 2 4 4 4

r2 1 3 2 2 2

r3 1 3 3 2

r4 1 1 3

r5 2 3

r6 1

Table 1

This completes the proof. 2References

[1] I.Cahit, Cordial graphs:A weaker version of graceful and harmonious graphs, Ars Combi-

natoria, 23(1987), 201-207.

[2] J.A.Gallian, A Dynamic survey of graph labeling, The Electronic Journal of Combinatorics,

19 (2017) #Ds6.

[3] F.Harary, Graph Theory, Addision wesley, New Delhi (1969).

[4] R.Ponraj, J.Maruthamani and R.Kala, k-Total prime cordial labeling of graphs, Journal

of Algorithms and Computation, 50(1), 143-149.

[5] R.Ponraj, J.Maruthamani and R.Kala, Some 4-total prime cordial labeling of graphs, Jour-

nal of Applied Mathematics and Informatics, 37(1-2)(2019), 149-156.

[6] R.Ponraj, J.Maruthamani and R.Kala, Some classes of 4-total prime cordial labeling of

graphs, Global Journal of Engineering science and Researches, 5(11), 319-327.

[7] R.Ponraj, J.Maruthamani and R.Kala, 4-total prime cordial labeling of some special graphs,

Jordan Journal of Mathematics and Statistics, (Accepted for Publication).

[8] R.Ponraj, J.Maruthamani and R.Kala, 4-Total prime cordial labeling of some cycle related

graphs, Journal of Algorithms and Computation, 50(2), 49-57.

[9] R.Ponraj and J.Maruthamani, 4-total prime cordial labeling of some derived graphs, Ars

Combinatoria, (Accepted for Publication).


The Signed Product Cordial for Corona of Paths and

Fourth Power of Paths

S. Nada, A. Elrokh

(Department of Mathematics, Faculty of Science, Menoyfia University, Egypt)

A. Elrayes and A. Rabie

(Institute of National Planning, Cairo, Egypt)

E-mail: [email protected] [email protected], [email protected] and [email protected]

Abstract: A graph G = (V, E) is called signed product cordial if it is possible to label the

vertex by the function f : V → {−1, 1} and label the edges by f∗ : E → {−1, 1}, where

f∗(uv) = f(u) · f(v), u, v ∈ V so that |v−1 − v1| 6 1 and |e−1 − e1| 6 1. In [3] J.Devaraj and

P.Delphy, they have defined signed graphs, and they have started by labeling edges and then

induced the labeling of vertices. In this paper, we contribute some new results on signed

product cordial labeling and present necessary and sufficient conditions for signed product

cordial for corona of paths and fourth power of paths.

Key Words: Second power, fourth power, corona graph, signed product cordial graph,

Smarandachely signed product cordial labeling.

AMS(2010): 05C78,05C75,05C20.

§1. Introduction

The labeling of graphs is perceived to be a primarily theoretical subject in the field of graph the-

ory and discrete mathematics, it serves as models in a wide range of application like astronomy,

coding theory, circuit design and communication networks addressing. The concept of graph

labeling was introduced during the sixties’ of the last century by Rosa [12]. Many researches

have been working with different types of labeling graphs [1,4,5]. In 1954 Harray introduced

S-cordiality [10]. An excellent reference for this purpose is the survey written by Gallian [6].

All graphs considered in this theme are finite, simple and undirected. The original concept of

cordial graphs is due to Chait[2]. He showed that each tree is cordial; an Euerlian graph is not

cordial if its size is congruent to 2(mod 4). Let G = (V,E) be a graph and let f : V → {−1, 1}be a labeling of its vertices, and let the induced edge labeling f∗ : E → {−1, 1} be given by

f∗(e) = (f(u) · f(v)), where e = uv and u, v ∈ V .

Let v−1 and v1 be the numbers of vertices that are labeled by −1 and 1, respectively, and

let e−1 and e1 be the corresponding numbers of edges. Such a labeling is called signed product

1Received April 14, 2019, Accepted December 6, 2019.

The Signed Product Cordial for Corona of Paths and Fourth Power of Paths 103

cordial if both |v−1 − v1| 6 1 and |e−1 − e1| 6 1 hold. Otherwise, it is called Smarandachely

signed product cordial if |v−1 − v1| > 1 or |e−1 − e1| > 1. The corona G1

⊙G2 of two graphs

G1 (with n1 vertices, m1 edges) and G2 (with n2 vertices, m2 edges) is defined as the graph

obtained by taking one copy of G1 and copies of G2, and then joining the ith vertex of G1 with

an edge to every vertex in the ith copy of G2. It is easy to see that the corona G1

⊙G2 that

has n1 + n1n2 vertices and m1 + n1m2 + n1n2 edges. The fourth power of a paths Pn, denoted

by P 4n , is Pn

⋃J , where J is the set of all edges of the form edges vivj such that 2 ≤ d(vivj) ≤ 4

and i < j where d(vivj) is the shortest path from vi to vj .

§2. Terminologies and Notations

A path with m vertices and m−1 edges, denoted by Pm, and its fourth power P 4n has n vertices

and 4n − 10 edges. We let L4r denote the labeling (−1)211 (−1)211· · · (−1)211 (repeated r-

times), Let L′4r denote the labeling (−1)11(−1) (−1)11(−1)· · · (−1)11(−1) (repeated r-times).

The labeling 11(−1)2 11(−1)2· · · 11(−1)2 (repeated r-times) and labeling 1(−1)211(−1)21

· · · 1(−1)21 (repeated r-times) are written S4r and S′4r. Let Mr denote the labeling (−1)1

(−1)1 · · · (−1)1, zero-one repeated rtimes if r is even and (−1)1 (−1)1 · · · (−1)1(−1) if r is

odd; for example, M6 = (−1)1(−1)1(−1)1 and M5 = (−1)1(−1)1(−1). We let M ′r denote the

labeling 1(−1)1(−1) · · · 1(−1). Sometimes, we modify the labeling Mr or M ′r by adding symbols

at one end or the other (or both). Also, L4r (or L′4r ) with extra labeling from right or left (or

both sides). If L is a labeling for a path pm and M is a labeling for fourth power of path Pn,

then we use the notation [L;M ] to represent the labeling of the corona Pm

⊙P 4

n . Additional

notation that we use is the following: for a given labeling of the corona Pm

⊙P 4

n , we let vi

and ei (for i = −1, 1) be the numbers of vertices and edges, respectively, that are labeled by i

of the corona Pm

⊙P 4

n , and let xi and ai be the corresponding quantities for pm, and we let

yi and bi be those for P 4n , which are connected with vertices labeled (−1) of Pm. Similarly, let

y′i and b′i for P 4n which are connected with vertices labeled 1 of Pm. It is easy to verify that

v−1 = x−1+x−1y−1+x1y′−1, v1 = x1+x−1y1+x1y

′1, e−1 = a−1+x−1b−1+x1b

′−1+x−1y−1+x1y

′1

and e1 = a1+x−1b1+x1b′1+x−1(x−1y1)+x1y

′−1. Thus, v−1−v1 = (x−1−x1)+x−1(y−1−y1)+

x1(y′−1−y′1) and e−1−e1 = (a−1−a1)+x−1(b−1−b1)+x1(b

′−1−b′1)+x−1(y−1−y1)−x1(y

′−1−y′1).

When it comes to the proof, we only need to show that, for each specified combination of

labeling, |v−1 − v1| ≤ 1 and |e−1 − e1| ≤ 1.

§3. Main Results

In this section, we show that the corona Pm

⊙P 4

n is signed product cordial for all m, n ≥ 7 and

also for n = 3 with m ≥ 1. This target will be achieved after the following series of lemmas.

Lemma 3.1 The corona Pm

⊙P 4

3 is signed product cordial if and only if m 6= 1.

Proof Suppose that n = 3. The following cases will be examined.

104 S. Nada, A. Elrokh, A. Elrayes and A. Rabie

Case 1. Obviously, P1

⊙P 4

3 isomorphic to the complete graph K4. Since K4 is not cordial,

P1

⊙P 4

3 is not is signed product cordial.

Case 2. Suppose that m = 2. Then we label the vertices of P2

⊙P 4

3 by [−11;−11−1, 1−11];

hence v−1 − v1 = 0 and e−1 − e1 = 1. So, P2

⊙P 4

3 is signed product cordial.


⊙P 4

3 by [−1 − 1 −1;−111, 111,−11 − 1]; hence v−1 − v1 = 0 and e−1 − e1 = 0. So, P3

⊙P 4

3 is signed prod-

uct cordial.

Case 4. m ≡ 0(mod4).

Suppose that m = 4r, r ≥ 1. We choose the labeling [L4r;−11 − 1,−11 − 1, 1 − 11, 1 −11, · · · , (r − times)] for P4r

⊙P 4

3 . Therefore x−1 = x1 = 2r, a−1 = 2r − 1, a1 = 2r, y−1 = 2,

y1 = 1, y′−1 = 1, y′1 = 2, b−1 = 2, b′−1 = 2, b1 = 1 and b′1 = 1. Hence v−1 − v1 =

(x−1 − x1) + x−1.(y−1 − y1) + x1.(y′−1 − y′1) = 0 and e−1 − e1 = (a−1 − a1) + x−1.(b−1 − b1) +

x1.(b′−1 − b′1) + x−1.(y−1 − y1)− x1.(y

′−1 − y′1) = −1. Thus P4r

⊙P 4

3 is signed product cordial.


Suppose that m = 4r+1, r ≥ 1. We choose the labeling [L4r1;−11−1, −11−1, 1−11, 1−11, · · · , (r − times),−11 − 1] for P4r+1

⊙P 4

3 . Therefore x−1 = 2r, x1 = 2r + 1, a−1 = 2r − 1,

a1 = 2r + 1, and for the first 4r- vertices y−1 = 2, y1 = 1, y′−1 = 1, y′1 = 2, b−1 = b′−1 = 2

and b1 = b′1 = 1, and for the cycle c3 which is connected to last vertex in P4r+1, we have

y′′−1 = 2, y′′1 = 1, b′′−1 = 2 and b′′(1) = 1, where y′′i and b′′i are the numbers of vertices and

edges labeled by i in P 43 that is connected to the last vertex of P4r+1. It is easily to verify that

v−1 = x−1 + x−1.y−1 + (x1 − 1).y′−1 + y′′−1 = 8r + 2, v1 = x1 + x−1.y1 + (x1 − 1).y′1 + y′′1 =

8r + 2, e−1 = a−1 + x−1.b−1 + (x1 − 1).b′−1 + x−1.y−1 + (x1 − 1).y′1 + b′′−1 + 1 = 14r + 3 and

e1 = a1 + x−1.b1 + (x1 − 1).b′1 + x−1.y1 + (x1 − 1).y′−1 + b′′(1) + 2 = 14r + 3. It follows that

v−1 − v1 = 0 and e−1 − e1 = 0. Thus P4r+1

⊙P 4

3 is cordial.


Suppose that m = 4r+2, r ≥ 1. We choose the labeling [L4r1(−1); (−1)1(−1), (−1)1(−1),

1(−1)1, 1(−1)1, ..., (r − times), 1(−1)1, (−1)1(−1)] for P4r+2

⊙P 4

3 . Therefore x−1 = x1 =

2r + 1, a−1 = 2r, a1 = 2r + 1, y−1 = 2, y1 = 1, y′−1 = 1, y′1 = 2, b−1 = 2, b′−1 = 2,

b1 = 1 and b′1 = 1. Hence v−1 − v1 = (x−1 − x1) + x−1.(y−1 − y1) + x1.(y′−1 − y′1) = 0 and

e−1 − e1 = (a−1 − a1) + x−1.(b−1 − b1) + x1.(b′−1 − b′1) + x−1.(y−1 − y1) − x1.(y

′−1 − y′1) = −1.

Thus P4r+2

⊙P 4

3 is cordial.


Let m = 4r + 3, r ≥ 1. We choose the labeling [L4r1(−1)(−1); (−1)1(−1), (−1)1(−1),

1(−1)1, 1(−1)1, · · · , (r − times), 1(−1)1, (−1)1(−1), 1(−1)1] for P4r+3

⊙P 4

3 . Therefore x−1 =

2r+2, x1 = 2r+1, a−1 = 2r, a1 = 2r+2, and for the first 4r- vertices y−1 = 2, y1 = 1, y′−1 = 1,

y′1 = 2, b−1 = b′−1 = 2 and b1 = b′1 = 1, and for the cycle P 43 which is connected to last vertex of

P4r+3, we have y′′0 = 1, y′′1 = 2, b′′0 = 2 and b′′1 = 1, where y′′i and b′′i are the numbers of vertices

and edges labeled by i in P 43 that is connected to the last vertex of P4r+3. Similar to Case 2, we


conclud that v−1 − v1 = (x−1 −x1)+x−1.(y−1 − y1)+ (x1 − 1).(y′−1 − y′1)+ (y′′−1 − y′′1 ) = 0 and

e−1 − e1 = (a−1 − a1) + x−1.(b−1 − b1) + (x1 − 1).(b′−1 − b′1) + x−1.(y−1 − y1)− (x1 − 1).(y′−1 −y′1) + (c−1 − c1) − 1 = 0. Hence P4r+3

⊙P 4

3 is cordial. Thus the lemma is proved. 2Lemma 3.2 If n ≡ 0(mod4), then Pm

⊙P 4

n is signed product cordial for all m ≥ 1.

Proof Suppose that n = 4s, where s ≥ 2. The following cases will be examined.


⊙P 4

4s by [−1;−1L4s−4 − 112].

Therefore x−1 = 1, x1 = 0, a−1 = a1 = 0, y−1 = y1 = 2s, b−1 = b1 = 8s − 5. It follows that

v−1 − v1 = 1 and e−1 − e1 = 0. As an example, Figure 1 illustrates P1

⊙P 4

8 . Hence, P1

⊙P 4

4s

is signed product cordial.

-1 -1 -1 1 1 -1 1 1

-1

Figure 1


⊙P 4

4s by [−11;−1L4s−4 −112, 12L

′4s−4−12]. Therefore x−1 = x1 = 1, a−1 = 1, a1 = 0, y−1 = y1 = 2s, b−1 = b1 =

8s− 5, y′−1 = y′1 = 2s and b′−1 = b′1 = 8s− 5. It follows that v−1 − v1 = 0 and e−1 − e1 = 1. As

an example, Figure 2 illustrates P2

⊙P 4

8 . Hence, P2

⊙P 4

4s is signed product cordial.

-1 -1 -1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1

-1 1

Figure 2



⊙P 4

4s by [−121;−1L4s−4 −112,−1L4s−4 − 112, 12L

′4s−4−12]. Therefore x−1 = 2, x1 = 2, a−1 = a1 = 1, y−1 = y1 =

2s, b−1 = b1 = 8s− 5, y′−1 = y′1 = 2s and b′−1 = b′1 = 8s− 5. It follows that v0−1 − v1 = 1 and

e−1 − e1 = 0. As an example, Figure 3 illustrates P3

⊙P 4

8 . Hence, P3

⊙P 4

4s is signed product

cordial.

-1 -1 -1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1

-1 -1 1

-1 -1 -1 1 1 -1 1 1

Figure 3

Case 4. m = 0(mod 4).

Suppose thatm = 4r, where r ≥ 2. Then we label the vertices of P4r

⊙P 4

4s by [L4r;−1L4s−4−112,−1L4s−4 − 112, 12L

′4s−4−12,

12L′4s−4−12, ..., (r − time)]. Therefore x−1 = x1 = 2r, a−1 = 2r − 1, a1 = 2r, y−1 = y1 =

2s, b−1 = b1 = 8s− 5, y′−1 = y′1 = 2s and b′−1 = b′1 = 8s− 5. It follows that v−1 − v1 = 0 and

e−1−e1 = 1. As an example, Figure 4 illustrates P4

⊙P 4

8 . Hence, P4r

⊙P 4

4s is signed product

cordial.

-1 -1 -1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1

-1 -1 1 1

-1 -1 -1 1 1 -1 1 1

Figure 4

Case 5. m = 1(mod 4).

Suppose that m = 4r + 1, where r ≥ 1. Then we label the vertices of P4r+1

⊙P 4

4s by

[L4r−1;−1L4s−4−112,−1L4s−4−112, 12L′4s−4−12, 12L

′4s−4−12, · · · , (r−time),−1L4s−4−112].

Therefore x−1 = 2r+1, x1 = 2r, a−1 = a1 = 2r, y−1 = y1 = 2s, b−1 = b1 = 8s−5, y′−1 = y′1 = 2s


and b′−1 = b′1 = 8s− 5. It follows that v−1 − v1 = 1 and e−1 − e1 = 0. Hence, P4r+1

⊙P 4

4s is

signed product cordial.

Case 6. m = 2(mod 4).


⊙P 4

4s by

[L4r−11;−1L4s−4−112,−1L4s−4−112, 12L′4s−4−12, 12L

′4s−4−12, · · · , (r−time), −1L4s−4−112,

12L′4s−4−12]. Therefore x−1 = x1 = 2r + 1, a−1 = 2r + 1, a1 = 2r, y−1 = y1 = 2s, b−1 = b1 =

8s − 5, y′−1 = y′1 = 2s and b′−1 = b′1 = 8s − 5. It follows that v−1 − v1 = 0 and e−1 − e1 = 1.

Hence, P4r+2

⊙P 4

4s is signed product cordial.

Case 7 m = 3(mod 4).


⊙P 4

4s by

[L4r−121;−1L4s−4−112,−1L4s−4−112, 12L′4s−4−12,

12L′4s−4v2, · · · , (r − time), −1L4s−4−112,−1L4s−4−112, 12L

′4s−4−12]. Therefore x−1 = 2r +

2, x1 = 2r + 1, a−1 = a1 = 2r + 1, y−1 = y1 = 2s, b−1 = b1 = 8s − 5, y′−1 = y′1 = 2s and

b′−1 = b′1 = 8s− 5. It follows that v−1 − v1 = 1 and e−1 − e1 = 0. Hence, P4r+3

⊙P 4

4s is signed

product cordial. 2Lemma 3.3 If n ≡ 1(mod4), then Pm

⊙P 4

n is cordial for all m ≥ 1.

Proof Suppose that n = 4s+ 1, where s ≥ 2. The following cases will be examined.

Case 1. Suppose thatm = 1. Then we label the vertices of P1

⊙P 4

4s+1 by [−1; 12L′4s−4−11−1].

Therefore x−1 = 1, x1 = 0, a−1 = a1 = 0, y−1 = 2s, y1 = 2s+ 1, b−1 = b1 = 8s− 3 . It follows

that v−1 − v1 = 0 and e−1 − e1 = 1. Hence, P1

⊙P 4

4s+1 is signed product cordial.


⊙P 4

4s+1 by [−11;−12L4s−41−11,

12L′4s−4−11−1]. Therefore x−1 = x1 = 1, a−1 = 1, a1 = 0, y−1 = 2s + 1, y1 = 2s, b−1 = b1 =

8s − 3, y′−1 = 2s + 1, y′1 = 2s and b′−1 = b′1 = 8s − 3. It follows that v−1 − v1 = 0 and

e−1 − e1 = −1. Hence, P2

⊙P 4


Case 3. Let m = 3. Then we label the vertices of P3

⊙P 4

4s+1 by [−11−1;−12L4s−41−11,

12L′4s−4−11−1, 12L

′4s−4−11−1]. Therefore x−1 = 2, x1 = 1, a−1 = 2, a1 = 0, y−1 = 2s+1, y1 =

2s, b−1 = b1 = 8s − 3, y′−1 = 2s, y′1 = 2s + 1, b′−1 = b′1 = 8s − 3, y′′−1 = 2s, y′′1 = 2s + 1

andb′′−1 = b′′1 = 8s − 3. It follows that v−1 − v1 = 0 and e−1 − e1 = 1. Hence, P3

⊙P 4

4s+1 is


Case 4. m = 0(mod 4).

Suppose that m = 4r, where r ≥ 1. Then we label the vertices of P4r

⊙P 4

4s+1 by

[M4r;−12L4s−41−11, 12L′4s−4−11−1,−12L4s−41−11,

12L′4s−4−11−1, · · · , (r − time)]. Therefore x−1 = x1 = 2r, a−1 = 4r − 1, a1 = 0, y−1 =

2s + 1, y1 = 2s, b−1 = b1 = 8s − 3, y′−1 = 2s, y′1 = 2s + 1 and b′−1 = b1 = 8s − 3. It fol-

lows that v−1 − v1 = 0 and e0 − e1 = −1. Hence, P4r

⊙P 4


Case 5. m = 1(mod 4).


⊙P 4

4s+1 by


[M4r−1;−12L4s−41−11, 12L′4s−4−11−1,−12L4s−41−11, 12L

′4s−4−11−1, · · · , (r−time), 12L

′4s−4

−11−1]. Therefore x−1 = 2r + 1, x1 = 2r, a−1 = 4r, a1 = 0, y−1 = 2s + 1, y1 = 2s, b−1 = b1 =

8s− 3, y′−1 = 2s, y′1 = 2s+ 1, b′−1 = b′1 = 8s− 3, y′′−1 = 2s, y′′1 = 2s+ 1and b′′−1 = b′′1 = 8s− 3. It

follows that v−1 − v1 = 0 and e−1 − e1 = 1. Hence, P4r+1

⊙P 4


Case 6. m = 2(mod 4).

Letm = 4r+2, where r ≥ 1. Label the vertices of P4r+2

⊙P 4

4s+1 by [M4r+2;−12L4s−41−11,

12L′4s−4−11−1,−12L4s−41−11, 12L

′4s−4−11−1,−12L4s−41−11, 12L

′4s−4−11−1, · · · , (r−time)].

Therefore x−1 = x1 = 2r+1, a−1 = 4r+1, a1 = 0, y−1 = 2s+1, y1 = 2s, b−1 = b1 = 8s−3, y′−1 =

2s, y′1 = 2s+ 1 and b′−1 = b′1 = 8s− 3. It follows that v−1 − v1 = 0 and e0 − e1 = −1. Hence

P4r+2

⊙P 4


Case 7. m = 3(mod 4).

Suppose that m = 4r + 3, where r ≥ 1. We then label the vertices of P4r+3

⊙P 4

4s+1

by [M4r+2−1; −12L4s−41−11, 12L′4s−4−11−1, −12L4s−41−11, 12L

′4s−4−11−1, −12L4s−41−11,

12L′4s−4−11−1, · · · , (r − time), 12L

′4s−4−11−1]. Therefore x−1 = 2r + 2, x1 = 2r + 1, a−1 =

4r + 1, a1 = 0, y−1 = 2s + 1, y1 = 2s, b−1 = b1 = 8s − 3, y′0 = 2s, y′1 = 2s + 1, b′0 = b′1 =

8s − 3, y′′−1 = 2s, y′′1 = 2s + 1 and b′′−1 = b′′1 = 8s − 3. It follows that v−1 − v1 = 0 and

e−1 − e1 = 1. Hence, P4r+3

⊙P 4

4s+1 is signed- cordial. 2Lemma 3.4 If n ≡ 2(mod4), then Pm

⊙P 4

n is cordial for all m ≥ 1.



⊙P 4

4s+2 by [−1;−113−1s4s−4−1].

Therefore x−1 = 1, x1 = 0, a−1 = a1 = 0, y−1 = y1 = 2s+ 1, b−1 = b1 = 8s− 1. It follows that

v−1 − v1 = 1 and e−1 − e1 = 0. Hence, P1

⊙P 4


Case 2. Suppose that m = 2. Then, label the vertices of P2

⊙P 4

4s+2 by [−11;−113−1s4s−4−1,

−1l4s−4−113−1]. Therefore x−1 = x1 = 1, a−1 = 1, a1 = 0, y−1 = y1 = 2s + 1, b−1 = b1 =

8s−1, y′−1 = y′1 = 2s+1 and b′−1 = b′1 = 8s−1. It follows that v−1−v1 = 0 and e−1−e1 = −1.

Hence, P2

⊙P 4


Case 3. Let m = 3. Then we label the vertices of P3

⊙P 4

4s+2 by [−1−11; −113−1s4s−4−1,

−113−1s4s−4−1, −1l4s−4−113−1]. Therefore x−1 = 2, x1 = 1, a−1 = a1 = 1, y−1 = y1 =

2s+ 1, b−1 = b1 = 8s− 1, y′−1 = y′1 = 2s+ 1 and b′−1 = b′1 = 8s− 1. It follows that v−1 − v1 = 1

and e−1 − e1 = 0. Hence, P3

⊙P 4


Case 4. m = 0(mod 4).


⊙P 4

4s+2 by

[l4r;−113−1s4s−4−1,−113−1s4s−4−1,−1L4s−4−113−1,−1L4s−4−113−1, · · · , (r−time)]. There-

fore x−1 = x1 = 2r, a−1 = 2r−1, a1 = 2r, y−1 = y1 = 2s+1, b−1 = b1 = 8s−1, y′−1 = y′1 = 2s+1

and b′−1 = b1 = 8s− 1. It follows that v−1 − v1 = 0 and e−1 − e1 = −1. Hence, P4r

⊙P 4

4s+2 is


Case 5. m = 1(mod 4).



⊙P 4

4s+2 by

[l4r−1;−113−1s4s−4−1,−113−1s4s−4−1,

−1L4s−4−113−1,−1L4s−4−113−1, · · · , (r − time), −113−1s4s−4−1]. Therefore x−1 = 2r +

1, x1 = 2r, a−1 = a1 = 2r, y−1 = y1 = 2s + 1, b−1 = b1 = 8s − 1, y′−1 = y′1 = 2s + 1 and

b′−1 = b′1 = 8s − 1. It follows that v−1 − v1 = 1 and e−1 − e1 = 0. Hence, P4r+1

⊙P 4

4s+2 is

signed- cordial.

Case 6. m = 2(mod 4).


⊙P 4

4s+2 by

[l4r−11;−113−1s4s−4−1,−113−1s4s−4−1,

−1l4s−4−113−1,−1l4s−4−113−1, · · · , (r − time), −113−1s4s−4−1,−1L4s−4−113−1]. There-

fore x−1 = x1 = 2r + 1, a−1 = 2r + 1, a1 = 2r, y−1 = y1 = 2s + 1, b−1 = b1 = 8s − 1, y′−1 =

y′1 = 2s + 1 and b′−1 = b′1 = 8s − 1. It follows that v−1 − v1 = 0 and e−1 − e1 = 1. Hence,

P4r+2

⊙P 4


Case 7. m = 3(mod 4).


⊙P 4

4s+2

by [L4r−121;−113−1s4s−4−1, −113−1s4s−4−1, −1L4s−4−113−1, −1L4s−4−113−1, · · · , (r −time), −113−1s4s−4−1, −113−1s4s−4−1,−1L4s−4−113−1]. Therefore x−1 = 2r + 2, x1 =

2r + 1, a−1 = a1 = 2r + 1, y−1 = y1 = 2s + 1, b−1 = b1 = 8s − 1, y′−1 = y′1 = 2s + 1 and

b′−1 = b′1 = 8s − 1. It follows that v−1 − v1 = 1 and e−1 − e1 = 0. Hence, P4r+3

⊙P 4

4s+2 is

signed product cordial. 2Lemma 3.5 If n ≡ 3(mod4), then Pm

⊙P 4

n is signed product cordial for all m ≥ 1.



⊙P 4

4s+3 by [−1; 12s4s−1].

Therefore x−1 = 1, x1 = 0, a−1 = a1 = 0, y−1 = 2s + 1, y1 = 2s + 2, b−1 = b1 = 8s + 1 . It

follows that v−1 − v1 = 0 and e−1 − e1 = 1. Hence, P1

⊙P 4


Case 2. Suppose thatm = 2. Then, label the vertices of P2

⊙P 4

4s+3 by [−11;−121L4s, 12s4s−1].

Therefore x−1 = x1 = 1, a−1 = 1, a1 = 0, y−1 = 2s + 2, y1 = 2s+ 1, b−1 = b1 = 8s + 1, y′−1 =

2s+1, y′1 = 2s+2 and b′−1 = b′1 = 8s+1. It follows that v−1− v1 = 0 and e−1− e1 = 1. Hence,

P2

⊙P 4



⊙P 4

4s+3 by [−11−1; −121L4s,

12s4s−1, 12s4s−1]. Therefore x−1 = 2, x1 = 1, a−1 = 2, a1 = 0, y−1 = 2s+ 2, y1 = 2s+1, b−1 =

b1 = 8s + 1, y′−1 = 2s + 1, y′1 = 2s + 2, b′−1 = b′1 = 8s + 1, y′′−1 = 2s + 1, y′′1 = 2s + 2

andb′′−1 = b′′1 = 8s + 1. It follows that v−1 − v1 = 0 and e−1 − e1 = 1. Hence, P3

⊙P 4

4s+3 is


Case 4. m = 0(mod 4).


⊙P 4

4s+3 by

[M4r;−121L4s, 12s4s−1,−121L4s, 12s4s−1, · · · , (r − time)]. Therefore x−1 = x1 = 2r, a−1 =


4r − 1, a1 = 4r, y−1 = 2s + 2, y1 = 2s + 1, b−1 = b1 = 8s + 1, y′−1 = 2s + 1, y′1 = 2s + 2 and

b′−1 = b′1 = 8s + 1. It follows that v−1 − v1 = 0 and e−1 − e1 = −1. Hence, P4r

⊙P 4

4s+3 is


Case 5. m = 1(mod 4).


⊙P 4

4s+3

by [M4r−1;−121L4s, 12s4s−1,−121L4s, 12s4s−1, · · · , (r − time), 12s4s−1]. Therefore x−1 =

2r+1, x1 = 2r, a−1 = 4r, a1 = 0, y−1 = 2s+2, y1 = 2s+1, b−1 = b1 = 8s+1, y′−1 = 2s+1, y′1 =

2s + 2, b′−1 = b′1 = 8s + 1, y′′−1 = 2s + 1, y′′1 = 2s + 2and b′′−1 = b′′1 = 8s + 1. It follows that

v−1 − v1 = 0 and e−1 − e1 = 1. Hence, P4r+1

⊙P 4


Case 6. m = 2(mod 4).


⊙P 4

4s+3

by [M4r+2;−121L4s, 12s4s−1,−121L4s, 12s4s−1,−121L4s, 12s4s−1, · · · , (r − time)]. Therefore

x−1 = x1 = 2r + 1, a−1 = 4r + 1, a1 = 0, y−1 = 2s+ 2, y1 = 2s + 1, b−1 = b1 = 8s + 1, y′−1 =

2s + 1, y′1 = 2s + 2 and b′−1 = b′1 = 8s + 1. It follows that v−1 − v1 = 0 and e−1 − e1 = −1.

Hence, P4r+2

⊙P 4


Case 7. m = 3(mod 4).


⊙P 4

4s+3 by

[M4r+2−1;−121L4s, 12s4s−1,−121L4s, 12s4s−1,

−121L4s, 12s4s−1..., (r − time)

12s4s−1]. Therefore x−1 = 2r + 2, x1 = 2r + 1, a−1 = 4r + 2, a1 = 0, y−1 = 2s + 2, y1 =

2s+ 1, b−1 = b1 = 8s+ 1, y′−1 = 2s+ 1, y′1 = 2s+ 2, b′−1 = b′1 = 8s+ 1, y′′−1 = 2s+ 1, y′′1 = 2s+ 2

and b′′−1 = b′′1 = 8s+ 1. It follows that v−1 − v1 = 0 and e−1 − e1 = 1. Hence, P4r+3

⊙P 4

4s+3

is signed product cordial. 2As a consequence of all lemmas mentioned above we conclude that the signed product

cordial of the corona between paths and fourth power of paths is cordial for all m, n ≥ 7 and

n = 3 for all m ≥ 1.

Theorem 3.1 The corona between paths and fourth power of paths is signed product cordial

for all m, n ≥ 7 and also for n = 3 with m ≥ 1.

Proof The proof follows directly from Lemma 3.1 to Lemma 3.5. 2Acknowledgement. The authors are thankful to the anonymous referee for useful suggestions

and valuable comments.

References

[1] I. Cahit, Cordial graphs – A weaker version of graceful and harmonious graphs, Ars Com-

binatoria, 23(1987), 201-207.

[2] I. Cahit, On cordial and 3-eqtitable labeling of graphs, Utiliuies Math., 37(1990).


[3] J.Devaraj and P.Delphy, On signed cordial graph, Int. J. of Mathematical Sciences and

Applications, Vol.1, No.3, September (2011), 1156-1167.

[4] A .T. Diab, On cordial labeling of the second power of paths with other graphs, Ars

Combinatoria, 97A(2010), 327-343.

[5] A. Elrokh and A. Abd El-hay, The cordiality of cone and lemniscate graphs, MJMS (sub-

mitted).

[6] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combina-

torics, 17(2010), DS6.

[7] S. W. Golomb, How to number a graph, in Graph Theory and Computing, R.C. Read ed.,

Academic Press, New York, 1972, 23-37.

[8] R. L. Graham, and N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J.

Alg. Discrete Math., 1(1980), 382-404.

[9] F.Harray, On the notion of a signed graph, Michigan Math. J., 2 (1953-1954), 143-146.

[10] M.Hovey, A-cordial Graphs, Discrete Mathematics, 93 (1991), 183-194.

[11] S. Nada, A. Elrokh and A. Abd El-hay. The cordiality of the second power of some graphs,

CHZ.Math Journal(submitted).

[12] A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs(Internat

Symposium, Rome, July 1966), Gordon and Breach, N.Y. and Dunod Paris (1967), 349-

355.

[13] M. A. Seoud, A. T. Diab, and E. A. Elsahawi, On strongly-C harmonious, relatively prime,

odd graceful and cordial graphs, Proc. Math. Phys. Soc. Egypt, 73(1998), 33-55.

[14] T.Zaslausky, Signed graphs, Discrete applied Mathematics, 4(1982), 47-74.


On the Eccentric Sequence of Composite Graphs

Yaser Alizadeh

(Department of Mathematics, Hakim Sabzevari University, Sabzevar, Iran)


Abstract: The eccentric sequence of a graph is defined as list of eccentricity of its vertices.

Eccentric sequence of composite graphs under seven graph products: line graph, sum, carte-

sian product, disjunction, symmetric difference, lexicographic product and corona product

is investigated. Also some family of non vertex transitive graphs that are self centered are

determined as product of graphs. It is proved that for any positive integer d, there is an

infinite family of non-vertex transitive self centered graphs with diameter d. The relation

between total eccentricity of a tree and total eccentricity of its line graph is given.

Key Words: Eccentricity, eccentric sequence, graph product, self centered graph.

AMS(2010): 05C76.

§1. Introduction

We consider only simple connected graphs in this paper. Let G = (V (G), E(G)) be a graph and

u, v be two vertices of G. The distance between u and v, dG(u, v) (simply d(u, v)) is the length

of shortest path connecting u and v. For a vertex v ∈ V (G), the eccentricity of v, εG(v) is the

maximum distance from v to other vertices in G. The maximum and the minimum eccentricity

among all vertices of G are called diameter diam(G) and radius rad(G) of G respectively. The

center of G, C(G) is the set of vertices whose eccentricity is equal to rad(G). A graph G is

called self centered if all of its vertices have a same eccentricity. Let V (G) = {v1, v2, · · · , vn}be the vertex set of G. Eccentric sequence of G is the sequence ε1, ε2, · · · , εn where εi is

the eccentricity of vertex vi. The minimum eccentric sequence of G, es(G) is the sequence

{εt11 , ε

t22 , · · · , εtk

k } where ε1, ε2 · · · , εk are the different eccentricities of vertices and ti denotes

the number of vertices with eccentricity εi and more over εi+1 = εi + 1 for 1 6 i 6 k − 1.

Note that ε1 = rad(G) and εk = diam(G). Eccentric sequence is interesting since it provides

information on the vertex eccentricities and some structural properties of the graph such as

diameter, radius and variability of vertex eccentricities. Call a sequence of positive integer

eccentric if it is eccentric sequence of a graph. In a series of papers several properties of

eccentric sequences are studied. For instance see surveys [3, 7, 11, 14, 17]. Characterization of

eccentric sequences of graphs was first considered by Lesniak [11] who characterized sequences

which are the eccentricity sequences of trees.


On the Eccentric Sequence of Composite Graphs 113

Study of graph invariants specially topological indices under graph products is very in-

terested in mathematical literature. Some properties and application are reported in surveys

[1, 2, 4 C 6, 8 C 10, 12, 15, 18]. In this paper, we study the eccentric sequence of composite

graphs. We obtain explicit formulas of eccentric sequence for some graph product such as: line

graph, sum, cartesian product, disjunction, symmetric difference, lexicographic product and

corona product. Two important topological indices based on eccentricity of vertices are the

total eccentricity and eccentric connectivity index. The total eccentricity of a graph G, ξ(G) is

the sum of eccentricities of its vertices. Clearly, if

es(G) = {εt11 , · · · , εtk

k }

then,

ξ(G) =k∑

i=1

tiε(vi).

The eccentric connectivity index of graph G, ECI(G), introduced by Sharma et al. [16], is

defined as

ECI(G) =∑

v∈V (G)

ε(v)deg(v),

where deg(v) denotes degree of vertex v. These topological indices have been used as mathe-

matical models for the prediction of biological activities of diverse nature. The automorphism

group of G is denoted with Aut(G). A graph is called vertex transitive if for any pair of vertices

u and v, there is an automorphism α such that α(u) = v. It is known that an automorphism

of a graph preserve the distance function. It follows that vertex-transitive graphs are always

regular and self centered graph.

In this paper we construct an infinite family of non-vertex transitive graphs which are

self centered. In the rest of the section, some standard graph products are introduced, then

eccentric sequence of graphs under these graph products is verified. First, we start with line

graph. Line graph of G, L(G) is a graph which each vertex of L(G) is associated with an edge

of G and two vertices in L(G) is adjacent if and only if the corresponding edges of G have

an end vertex in common. The sum of two graphs G1 and G2, G1 + G2 is defined as the

graph with the vertex set V (G1) ∪ V (G2) and the edge set E (G1 +G2) = E (G1) ∪ E (G2) ∪{u1u2 |u1 ∈ V (G1), u2 ∈ V (G2)} . The next binary graph product is cartesian product. The

Cartesian product G1�G2 is the graph with vertex set V (G1)× V (G2) and (u1, u2) is adjacent

to (v1, v2) if u1 = v1 and (u2v2) ∈ E(G2), or u2 = v2 and (u1v1) ∈ E(G1).

The disjunction G1 ∨G2 is the graph with vertex set V (G1) × V (G2) and

E (G1 ∨G2) = {(u1, u2)(v1, v2) |u1v1 ∈ E(G1) or u2v2 ∈ E(G2) } .

For given graphs G1 and G2, their symmetric difference G1

⊕G2 is the graph with vertex

set V (G1)×V (G2) and two vertices (u1, u2) and (v1, v2) are adjacent if and only if u1v1 ∈ E(G1)

or u2v2 ∈ E(G2)

The diameter of disjunction and symmetric difference of two graphs when both of them

114 Yaser Alizadeh

contain more than one vertex do not exceed of 2. The next binary operation is the lexicographic

product. The lexicographic product of two graphs G1 and G2, G1 [G2] is the graph with vertex

set V (G1) × V (G2) and two vertices u = (u1, u2) and v = (v1, v2) are adjacent if (u1 is

adjacent with v1) or (u1 = v1 and u2 and v2 are adjacent).The operations sum, disjunction and

symmetric difference are symmetric operation and this fact implies that they have symmetric

eccentric sequence. But the lexicographic product do not have such property. Let ni, i = 1, 2

denotes the order of Gi. The corona product of two graphs is denoted by G1◦G2 and is obtained

from one copy of G1 and n1 copies of G2, and then joining all vertices of the i-th copy of G2 to

the i-th vertex of G1 for i = 1, 2, · · · , n1. In [13], application of coronas in chemical modeling

was reported.

§2. Main Result

In this section, explicit formulas for eccentric sequence of some composite graphs is given.

2.1 Line Graph of Trees

Eccentric sequence of line graph of a tree can be determined by its eccentric sequence. We

present a relation between of total eccentricity of a tree and total eccentricity of its line graph.

Theorem 2.1 Let T be a tree with rad(T) = r. If es(T ) = {rn0 , (r + 1)n1 , · · · , (2r)nr}, then

eccentric sequence of L(T ) is obtained as

es(L(T )) =

{rn1 , (r + 1)n2 , · · · , (2r − 1)nr} if C(T ) = K1

{(r − 1)1, rn1 , · · · , (2r − 2)nr−1} if C(T ) = K2

where nr > 0 and for 0 6 i 6 r − 1, ni > 1.

Proof We must to consider two cases.

Case 1. C(T ) = K1.

Let p be the unique central vertex. Let Ni(p) = {v ∈ V (T )|d(p, v) = i}. Since T is a tree,

for a vertex u ∈ Ni(p), εT (u) = i+ r. Also there is a unique vertex v ∈ Ni−1(p) that is adjacent

to u. Now consider the bijection f : V (T ) − {p} → E(T ), if u ∈ Ni(p), i > 1, then f(u) = uv

where v ∈ Ni−1(p). For a vertex u ∈ Ni(p), we have εT (u) = r+ i and εL(T )(f(u)) = r+ i− 1.

Thus if

es(T ) = {r1, (r + 1)n1 , · · · , (2r)nr},

the eccentric sequence of L(T ) is obtained as

es(L(T )) = {rn1 , (r + 1)n2 , . . . , (2r − 1)nr}.

Case 2. C(T ) = K2.


Let p1 and p2 be two adjacent central vertices of T . It is easy to see εL(T )(p1p2) = r − 1.

Let T−{p1p2} = T1∪T2 which pj ∈ Tj for j = 1, 2. Clearly if u ∈ Tj, clearly ε(u) = r+d(u, pj).

By a similar argument to Case 1, if u ∈ Ni(Pj)∩ Tj , j = 1, 2, then there is a unique vertex v ∈Ni−1(Pj) that uv ∈ E(T ). Thus we get again a bijection f : V (T )−{p1, p2} → E(T )−{p1p2}.Also if u ∈ Ni(Pj) ∩ Tj, then εL(T )(f(u)) = εT (u) − 1. This implies that if

es(T ) = {r2, (r + 1)n1 , · · · , (2r − 1)nr−1},

then

es(L(T )) = {(r − 1)1, (r)n1 , · · · , (2r − 2)nr−1}. 2Corollary 2.2 Let T be a tree. L(T ) is self centered graph if and only if T is a star graph.

Proof Clearly the line graph of a star graph is complete graph and then is self centered.

Let T be a tree of order n and rad(T) = r. If L(T ) is self center graph, by Theorem 2.1,

eccentric sequence of T has form es(T ) = {(r − 1)n−1} or {r1, (r + 1)n−2}. This means that

the variability of eccentricity in T is 1 or 2. Since T is a tree, this implies that rad(T) = 1 and

the proof is completed. 2Corollary 2.3 Let T be a tree of order n and radius r. Then

ξ(T ) = ξ(L(T )) + n+ r − 1.

Proof We consider two cases with respect to es(T ) and es(L(T )).

Case 1. es(T ) = {r1, (r + 1)n1 , · · · , (2r)nr} and es(L(T )) = {rn1 , (r + 1)n2 , · · · , (2r − 1)nr}.

By a straight calculation we get

ξ(T ) − ξ(L(T )) = r +

r∑

i=1

ni = r + n− 1.

Case 2. es(T ) = {r2, (r+1)n1 , · · · , (2r−1)nr−1} and es(L(T )) = {r − 11, (r)n1 , · · · , (2r−2)nr−1}.

Again, in this case a same result is obtained as well.

ξ(T ) − ξ(L(T )) = 2r − (r − 1) +

r−1∑

i=1

ni = r + 1 + n− 2 = n+ r − 1. 22.2 Sum

Theorem 2.4 Let G1 and G2 be simple connected graphs. Then,

es(G1 +G2) = {1c1+c2 , 2n1+n2−c1−c2},

116 Yaser Alizadeh

where ci is the number of vertices of eccentricity 1 in Gi and ni is the number of vertices of

Gi; i = 1, 2.

Proof It is not difficult to see that diam(G1 + G2) 6 2. For vertex x ∈ V (Gi) we have

εG1+G2(x) = 1 if and only if εGi(x) = 1. Let ci denotes the number vertices of eccentricity

1 in Gi, i = 1, 2. Then the eccentric sequence of G1 + G2 is obtained as es(G1 + G2) =

{1c1+c2 , 2n1+n2−c1−c2}. 2The eccentric sequence of sum of more than two graphs can be obtained by a reasoning

similar to the above.

Corollary 2.5 Let G1, G2, · · · , Gk be simple connected graphs. Then

es(G1 +G2 + · · · +Gk) = {1∑

ki=1 ci , 2

∑ki=1 ni−ci},

where ci is the number of vertices of eccentricity 1 (or 0) in Gi and ni is the number of vertices

of Gi; i = 1, 2, · · ·n.

Corollary 2.6 For any integer n > 5, there is a self centered graph and non vertex transitive

of diameter 2 and order n.

Proof It is sufficient to consider the complete bipartite graph K2,n−2 = K2 + Kn−2. 22.3 Cartesian Product

Theorem 2.7 Let es(G1) = {ε1t1 , ε2t2 , · · · , εk

tk} and es(G2) = {δ1s1 , δ2s2 , · · · , δmsm}.

Then,

es(G1�G2) ={

(εi + δj)tisj

}

16i6k, 16j6m.

Proof It is known that dG1�G2((x, y), (u, v)) = dG1(x, u) + dG2(y, v), this implies that

εG1�G2(x, y) = εG1(x) + εG2(y).

Let mi and nj be the number of vertices of eccentricity εi and δj in G1 and G2 re-

spectively. Then, minj vertices of G1�G2 have eccentricity εi + δj . Therefore es(G1�G2) ={(εi + δj)

tisj

}

16i6k, 16j6m. 2

Corollary 2.8 There are infinite family of non-vertex transitive self centered graph.

Proof It is sufficient to consider the powers of a non-vertex transitive self centered graph

such as Km,n where m 6= n and m,n > 2. 22.4 Disjunction

First note that if G = K1 then G ∨H ∼= H and G⊕H ∼= H as well. Therefore the considered

graph for these two graph products are except K1.


Theorem 2.9 Let G1 6= K1 6= G2. Then

es(G1 ∨G2) = {1c1c2 , 2n1n2−c1c2}

where ci is the number of vertices of eccentricity 1 in Gi and ni is the number of vertices of

Gi; i = 1, 2.

Proof Let (x, y) and (u, v) be two vertices of G1 ∨G2 and xx′ ∈ E(G1) and vv′ ∈ E(G2).

Since (x, y) and (u, v) both are adjacent to (x′, v′) then d((x, y), (u, v)) 6 2. Therefore for each

vertex (x, y) ∈ G1 ∨ G2, we have εG1∨G2(x, y) 6 2. If εG1(x) > 1, and dG1(x, y) > 2 then

d((x, u), (y, u)) = 2. Hence, εG1∨G2(x, y) = 1 if and only if εG1(x) = 1 = εG2(y). Let ci vertices

of Gi are of eccentricity 1 for i = 1, 2. Then c1c2 vertices of G1 ∨G2 have eccentricity 1 and the

other vertices are of eccentricity 2. This implies that es(G1 ∨G2) = {1c1c2 , 2n1n2−c1c2}. 2Corollary 2.10 Let G1 and G2 be two graphs of radius at least 2. Then G1 ∨ G2 is a self

centered graph and es(G1 ∨G2) = {2n1n2}.

2.5 Symmetric Difference

Lemma 2.11([9]) Let G1 and G2 be two simple connected graphs. The number of vertices

of Gi is denoted by ni for i = 1, 2. Then degG1⊕G2((u, v)) = n2degG1(u) + n1degG2(v) −2degG1(u)degG2(v).

Theorem 2.12 Let G1 6= K1 6= G2. Then es(G1 ⊕G2) = {2n1n2}.

Proof Let (x, y) and (u, v) be two vertices of G1 ∨G2 and xx′ ∈ E(G1) and vv′ ∈ E(G2).

Since (x, y) and (u, v) are adjacent to (x, v′) then d((x, y), (u, v)) 6 2. On the other hand,

(x, y) and (x′, v′) are not adjacent in G1 ⊕G2. Therefore each vertex (x, y) ∈ G1 ⊕G2 we have

εG1⊕G2(x, y) = 2. This concludes that es(G1 ⊕G2) = {2n1n2}. 2Corollary 2.13 For any positive integer d there is an infinite family of non vertex transitive

graphs which are self centered with diameter d.

Proof Let ⊕ki=1Gi = G1 ⊕ G2 ⊕ · · · ⊕ Gk. For any positive integers n, k > 3, let Gn,k =

⊕ki=1Pn. Using Lemma 2.11 and Theorem 2.12 we get that Gn,k is a self centered graph of

diameter 2 and it is non vertex transitive because it is not regular. Now consider the graph

Hn,k,d = Gn,k�C2(d−2) which is self center graph of diameter d. Since Gn,k is not regular

graph then Hn,k,d is not regular and consequently is not vertex transitive graph as well. Clearly

diameter of Hn,k,d is d and the proof is completed. 22.6 Lexicographic Product

For the lexicographic product of graphs, the distance of pair vertices is determined by the

following lemma.

118 Yaser Alizadeh

Lemma 2.14([9]) The distance of pair vertices in G1[G2] is

dG1[G2] ((u1, v1), (u2, v2)) =

dG1(u1, u2) v1 = v2

1 u1 = u2, v1v2 ∈ E(G2)

2 otherwise

Therefore, the eccentricity of vertex (u, v) ∈ V (G1[G2]) is determined by

ε(u, v) =

1 if εG1(u) = εG2(v) = 1

2 if εG1(u) = 1 and εG2(v) > 2

εG1(u) if ε(u) > 2

Now, all conditions are ready to obtain the eccentric sequence of G1[G2].

Theorem 2.15 Let es(G1) = {εt11 , . . . , ε

tk

k } and ni and ci, i = 1, 2 be the order and the number

of vertices of eccentricity 1 of Gi respectively. Then

es(G1[G2]) = {1c1c2 , 2c1(n2−c2)+t2n2 , εt3n23 , · · · , εtkn2

k }.

2.7 Corona

Remark 2.16 If G1 = K1, then G1oG2 = K1 +G2 and

es(G1oG2) = {11+c2 , 2n2−c2}

.

Theorem 2.17 Let G1 6= K1 and es(G1) ={εti

i

}k

i=1. Then

es(G1oG2) ={εt12 , ε

t2+n2t13 , εt3+n2t2

4 , · · · , εtk+n2tk−1

k+1 , εn2tk

k+2

},

where n2 = |V (G2)| and εi+1 = εi + 1.

Proof Let V (G1) = {v1, v2, . . . , vn} and G2,i be the copy of G2 associated to vi. From

the structure of corona product of graphs one can see that εG1o G2(vi) = εG1(vi) + 1 and if

x ∈ V (G2,i), εG1oG2(x) = εG1(vi) + 2, 1 6 i 6 n. Then for x ∈ V (G1oG2),

ε2 = ε1 + 1 6 ε(x) 6 εk + 2 = εk+2.

Let v be a central vertex of G1 and εG1(v) = ε1, then εG1oG2(v) = ε1 +1 = ε2. This follows that

center of G1 coincides center of G1oG2. For i > 2, the set of vertices of G1 having eccentricity

εi and the vertices of G2,t which ε(Vt) = εi−1 are of eccentricity εi+1 in G1oG2. Thus for i > 2,

the number of vertices that have eccentricity εi+1 is ti + n2ti−1. The proof is completed. 2


§3. Examples and Concluding Remarks

In this section, our theorems for eccentric sequence are illustrated for several more particular

composite graphs. We first give the expressions for suspensions.

Corollary 2.18 Let G be a graph on n vertices. Then

es(K1 +G) = {1c+1, 2n−c},

where c is the number of vertices of eccentricity 1 in G.

Next, the eccentric sequence for the fan graph K1 +Pn and the wheel graph Wn = K1 +Cn

are presented by

Corollary 2.19 For the fan graph K1 + Pn and the wheel graph Wn = K1 + Cn,

es(K1 + Pn) =

{12} if n = 1,

{13} if n = 2,

{12, 22} if n = 3,

{11, 2n} if n > 4

and

es(Wn) =

{14} if n = 3

{11, 2n} if n > 4.

By composing paths and cycles with various small graphs, we can obtain different classes

of polymer like graphs. For example, we state the eccentric sequence for the fence graph Pn[K2]

and the closed fence Cn[K2] in the following conclusion.

Corollary 2.20 For the fence graph Pn[K2] and the closed fence Cn[K2],

es(Pn[K2]) =

{12} if n = 1,

{14} if n = 2,

{12, 24} if n = 3,

{22t2 , 32t3 , · · ·k2tk} if n > 4,

where es(Pn) = {2t2, 3t3 , · · · ktk} for n > 4 and

es(Cn[K2]) =

{16} if n = 3,

{[n2 ]2n} if n > 4.

The t-thorny graph of a given graph G is obtained as GoKn, where Kn denotes the empty

graph on n vertices. For the t-thorny path and t-thorny cycle we get the following eccentric

120 Yaser Alizadeh

sequence.

Corollary 2.21 For the t-thorny graph PnoKt,

es(PnoKt) =

{11, 2t} n = 1,

{22, 32t} n = 2,

{21, 3t+2, 42t} n = 3.

If n > 4 and it is even

es(PnoKt) = {(n2

+ 1)2, (n

2+ 2)2t+2, · · · , n2t+2, (n+ 1)2t}

and if n > 5 and it is odd then

es(PnoKt) = {(n+ 1

2)1, (

n+ 1

2+ 1)t+2, (

n+ 1

2+ 2)2t+2, · · · , n2t+2, (n+ 1)2t}.

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Notes of New York, 37 (1999), 20-24.

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Chem., 5 (2014), 35–44.


F -Centroidal Mean Labeling of Graphs Obtained From Paths

S. Arockiaraj1, A. Rajesh Kannan2 and A. Durai Baskar3

1. Department of Mathematics, Government Arts & Science College, Sivakasi- 626124, Tamil Nadu, India

2. Department of Mathematics, Mepco Schlenk Engineering College, Sivakasi- 626005, Tamil Nadu, India

3. Department of Mathematics, C.S.I. Jayaraj Annapackiam College, Nallur - 627853, Tamil Nadu, India


Abstract: A function f is called an F -centroidal mean labeling of a graph G(V, E) with p

vertices and q edges if f : V (G) → {1, 2, 3, · · · , q + 1} is injective and the induced function

f∗ : E(G) → {1, 2, 3, · · · , q} defined as

f∗(uv) =

⌊2 [f(u)2 + f(u)f(v) + f(v)2]

3 [f(u) + f(v)]

⌋,

for all uv ∈ E(G), is bijective. A graph that admits an F -centroidal mean labeling is called

an F -centroidal mean graph. In this paper, we have discussed the F -centroidal meanness of

the graph Pn(X1, X2, · · ·Xn), the twig graph TW (Pn), the graph Pn ◦Sm for m ≤ 4, planar

grid Pm × Pn for m ≤ 3, the ladder graph Ln, the graph Pn ◦ K2, the graph P ba for a ≥ 2

and b ≤ 3, the middle graph of the path, total graph of the path and the square graph of

the path, the splitting graph of the path and the graph P (1, 2, · · · , n − 1).

Key Words: Labeling, F -centroidal mean labeling, F -centroidal mean graph, Smaran-

dachely F -centroidal mean labeling.

AMS(2010): 05C78.

§1. Introduction

Throughout this paper, by a graph we mean a finite, undirected and simple graph. Let G(V,E)

be a graph with p vertices and q edges. For notations and terminology, we follow [8]. For a

detailed survey on graph labeling, we refer [7].

Path on n vertices is denoted by Pn. The graph Pn(X1, X2, . . . Xn), is a tree obtained

from a path on n vertices by attaching Xi pendent vertices at each ith vertex of the path, for

1 ≤ i ≤ n. A Twig TW (Pn), n ≥ 4 is a graph obtained from a path by attaching exactly two

pendant vertices to each internal vertices of the path Pn. The graph G ◦Sm is obtained from G

by attaching m pendant vertices to each vertex of G. Let G1 and G2 be any two graphs with

p1 and p2 vertices respectively. Then the Cartesian product G1 × G2 has p1p2 vertices which

are {(u, v) : u ∈ G1, v ∈ G2} and the edges are obtained as follows: (u1, v1) and (u2, v2) are

adjacent in G1×G2 if either u1 = u2 and v1 and v2 are adjacent in G2 or u1 and u2 are adjacent

1Received January 11, 2019, Accepted December 9, 2019.

F -Centroidal Mean Labeling of Graphs Obtained From Paths 123

in G1 and v1 = v2. The product Pm × Pn and is called a planar grid and P2 × Pn is called a

ladder, denoted by Ln.

Let a and b be integers such that a ≥ 2 and b ≥ 2. Let y1, y2, . . . ya be the ′a′ fixed

vertices. Connect yi and yi+1 by means of b internally disjoint paths P ji of length ′i+ 1′ each,

for 1 ≤ i ≤ a− 1 and 1 ≤ j ≤ b. The resulting graph embedded in the plane is denoted by P ba .

The middle graph M(G) of a graph G is the graph whose vertex set is {v : v ∈ V (G)} ∪ {e :

e ∈ E(G)} and the edge set is {e1e2 : e1, e2 ∈ E(G) and e1 and e2 are adjacent edges of

G} ∪ {ve : v ∈ V (G), e ∈ E(G) and e is incident with v}. The total graph T (G) of a graph G is

the graph whose vertex set is V (G) ∪ E(G) and two vertices are adjacent if and only if either

they are adjacent vertices of G or adjacent edges of G or one is a vertex of G and the other

one is an edge incident on it. Square of a graph G, denoted by G2, has the vertex set as in G

and two vertices are adjacent in G2 if they are at a distance either 1 or 2 apart in G. For each

vertex v of the graph G, take a new vertex v′ to these vertices of G adjacent to v. The graph

thus obtained is called the splitting graph G and it is denoted by S′(G). An arbitrary super

subdivision P (m1,m2, · · · ,mn−1) of a path Pn is a graph obtained by replacing each ith edge

of Pn by identifying its end vertices of the edge with a partition of K2,mihaving 2 elements,

where mi is any positive integer.

Durai Baskar and Arockiaraj defined the F -harmonic mean labeling [6] and discussed its

meanness of some standard graphs. The concept of F -geometric mean labeling was introduced

by Durai Baskar and Arockiaraj [5] and it was developed [4]. The concept of F -root square

mean labeling was introduced by Arockiaraj et al., [1] and they studied the F -root square

mean labeling of some standard graphs [2]. Durai Baskar and Manivannan were introduced

F -heronian mean labeling [3]. Motivated by the works of so many authors in the area of graph

labeling, we introduced a new type of labeling called an F -centroidal mean labeling.

A function f is called an F -centroidal mean labeling of a graph G(V,E) with p vertices

and q edges if f : V (G) → {1, 2, 3, · · · , q+1} is injective and the induced function f∗ : E(G) →{1, 2, 3, · · · , q} defined by

f∗(uv) =

⌊2 [f(u)2 + f(u)f(v) + f(v)2]

3 [f(u) + f(v)]

⌋,

for all uv ∈ E(G), is bijective. Otherwise, it is called a Smarandachely F -centroidal mean

labeling of G if there is a number k ∈ {1, 2, 3, · · · , q} such that the inverse f−∗ of f∗ holds with

|f−∗(k)| ≥ 2. A graph that admits an F -centroidal mean labeling is called an F -centroidal

mean graph.

An F -centroidal mean labeling of cycle C4 is given in Figure 1.rrr r2

1

1 2 4

3 5

4

Figure 1 An F -centroidal mean labeling labeling of C4

124 S. Arockiaraj, A. Rajesh Kannan and A. Durai Baskar

In this paper, we have discussed the F -centroidal meanness of the graph Pn(X1, X2, · · ·Xn),

the twig graph TW (Pn), the graph Pn ◦ Sm for m ≤ 4, planar grid Pm × Pn for m ≤ 3, the

ladder graph Ln, the graph Pn ◦K2, the graph P ba for a ≥ 2 and b ≤ 3, the middle graph of the

path, total graph of the path and the square graph of the path, the splitting graph of the path

and the graph P (1, 2, · · · , n− 1).

§2. Main Results

Theorem 2.1 The graph Pn(X1, X2, · · ·Xn) is an F -centroidal mean graph, for 1 ≤ Xi ≤ 3

and |Xi −Xi+1| ≤ 1, for 1 ≤ i ≤ n− 1.

Proof Let u1, u2, · · · , un be the vertices of the path Pn. Let v(1)i , v

(2)i , · · · , v(Xi)

i be the

pendant vertices attached at ui, for 1 ≤ i ≤ n.

Define f : V (Pn(X1, X2, · · ·Xn)) → {1, 2, 3, · · · ,n∑

i=1

Xi + n} as follows:

f(v(1)i ) =

2, X1 = 1,

1, X1 6= 1.

For 2 ≤ i ≤ n,

f(v(1)i ) =

i−1∑k=1

Xk + i, Xi = 2, 3,

i−1∑k=1

Xk + i+ 1, Xi = 1.

For 1 ≤ i ≤ n,

f(v(j)i ) =

f(v

(1)i ) + 2, j = 2

f(v(1)i ) + 3, Xi = 3 and j = 3

and

f(ui) =

f(v(1)i ) + 1, Xi = 2, 3,

f(v(1)i ) − 1, Xi = 1.

Then the induced edge labeling f∗ is obtained as follows:

For 1 ≤ i ≤ n− 1,

f∗(uiui+1) =

f(ui) + 1, Xi = 1, 2,

f(ui) + 2, Xi = 3

and f∗(v(1)1 u1) = 1.


For 1 ≤ i ≤ n,

f∗(v(1)i ui) =

f(v

(1)i ) + 1, Xi = 2, 3,

f(v(1)i ) − 1, Xi = 1

and

f∗(v(j)i ui) =

f(ui), Xi = 2, 3 and j = 2,

f(ui) + 1, Xi = 3 and j = 3.

Hence f is an F -centroidal mean labeling of Pn(X1, X2, · · ·Xn).Thus the graph Pn(X1, X2, · · ·Xn)

is an F -centroidal mean graph, for 1 ≤ Xi ≤ 3 and |Xi −Xi+1| ≤ 1, for 1 ≤ i ≤ n− 1. 21 3 4 6 8 9 11 12 14 15 16 18 19 20 22

2 5 7 10 13 17 213 6 8 11 15 19

1 2 4 5 7 9 10 1213

14 16 18 20 2117

s s s s s sss s s s s s s s s s s s s s sFigure 2 An F -centroidal mean labeling of Pn(2, 2, 1, 2, 3, 3, 2)

Corollary 2.2 The twig graph TW (Pn) of the path Pn is an F -centroidal mean graph, for

n ≥ 4.

Theorem 2.3 The graph Pn ◦ Sm is an F -centroidal mean graph, for n ≥ 1 and m ≤ 4.

Proof Let v1, v2, v3, · · · , vn be the vertices of the path Pn and u(i)1 , u

(i)2 , u

(i)3 , · · · , u(i)

m be

the pendant vertices at each vi, for 1 ≤ i ≤ n.

Case 1. m = 4.

Define f : V (Pn ◦ S4) → {1, 2, 3, . . . , 5n} as follows:

f(v1) = 2,

f(vi) = 5i− 2, for 1 ≤ i ≤ n,

f(u(1)1 ) = 1,

f(u(i)1 ) = 5i− 5, for 2 ≤ i ≤ n,

f(u(1)2 ) = 3,

f(u(i)2 ) = 5i− 3, for 2 ≤ i ≤ n,

f(u(i)3 ) = 5i− 1, for 1 ≤ i ≤ n,

f(u(i)4 ) = 5i+ 1, for 1 ≤ i ≤ n− 1,

f(u(n)4 ) = 5n.



f∗(vivi+1) = 5i, for 1 ≤ i ≤ n− 1,

f∗(viu(i)1 ) = 5i− 4, for 1 ≤ i ≤ n,

f∗(viu(i)2 ) = 5i− 3, for 1 ≤ i ≤ n,

f∗(viu(i)3 ) = 5i− 2, for 1 ≤ i ≤ n

f∗(viu(i)4 ) = 5i− 1, for 1 ≤ i ≤ n.

Case 2. 1 ≤ m ≤ 3.

By Theorem 2.1, the results follows in this case.

Hence f is an F -centroidal mean labeling of Pn ◦Sm, for n ≥ 1 and m ≤ 4. Thus the graph

Pn ◦ Sm is an F -centroidal mean graph, for n ≥ 1 and m ≤ 4. 2sTTTTTs s ss TTTTTs s ss TTTTTs s ss TTTTTs s sss2 8 13 18

3 4 6 5 7 9 11 10 12 14 16 15 17 19 20

5 10 15

12 3

4 67 8

9 1112 13

1417

16 19

18

s s1

Figure 3 An F -centroidal mean labeling of P4 ◦ S4

Theorem 2.4 The planar grid Pm ×Pn, is an F -centroidal mean graph, for m ≤ 3 and n ≥ 2.

Proof Let V (Pm × Pn) = {vij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} and E(Pm × Pn) = {vijv(i+1)j :

1 ≤ i ≤ m− 1, 1 ≤ j ≤ n}∪ {vijvi(j+1) : 1 ≤ i ≤ m, 1 ≤ j ≤ n− 1} be the vertex set and edge

set of the graph Pm × Pn.

Case 1. m = 2.

Define f : V (P2 × Pn) → {1, 2, 3, · · · , 3n− 1} as follows:

f(vij) = i+ 3(j − 1), for 1 ≤ i ≤ 2 and 1 ≤ j ≤ n.


f∗(v1jv2j) = 3j − 2, for 1 ≤ i ≤ n,

f∗(vijvi(j+1)) = i+ 3j − 2, for 1 ≤ i ≤ 2 and 1 ≤ j ≤ n− 1.

Case 2. m = 3.


Define f : V (P3 × Pn) → {1, 2, 3, · · · , 5n− 2} as follows:

f(vi1) = i, for 1 ≤ i ≤ 3,

f(vi2) =

i+ 4, i = 1,

i+ 5, 2 ≤ i ≤ 3,

f(vij) = i+ 5(j − 1), for 1 ≤ i ≤ 3 and 3 ≤ j ≤ n.


f∗(vi1v(i+1)1) = i, for 1 ≤ i ≤ 2,

f∗(vi1vi2) = i+ 2, for 1 ≤ i ≤ 3,

f∗(vijv(i+1)j) =

2i+ 3(j − 1), 1 ≤ i ≤ 2 and j = 2,

i+ 5(j − 1), 1 ≤ i ≤ 2 and 3 ≤ j ≤ n,

f∗(vijvi(j+1)) = i+ 5j − 3, for 1 ≤ i ≤ 3 and 2 ≤ j ≤ n− 1.

Hence the graph Pm × Pn admits an F -centroidal mean labeling. Thus the graph Pm × Pn is

an F -centroidal meangraph for m ≤ 3.

For n = 2, an F -centroidal mean labeling of P2 × P4 is as shown in Figure 4. 2tttttttt12 4 5 7 8 10

10

11986532

1 4 7

Figure 4 An F -centroidal mean labeling of P2 × P4t t t ttt t t t ttt t t t ttt

51 11 16 21 26 31tt

3 8 13 18 23 28 33

27 12 17 22 27

32t13

6

4

2 75

8

11

12

9

10

1316

14

1517

1821

19

2022

2326

24

25

27

2831

29

30

32

Figure 5 An F -centroidal mean labeling of P3 × P7

Corollary 2.5 Every Ladder graph Ln = P2 × Pn is an F -centroidal mean graph for n ≥ 2.

Theorem 2.6 The graph Pn ◦K2 is an F -centroidal mean graph for n ≥ 1.


Proof Let v1, v2, v3, · · · , vn be the vertices of the path Pn and u(1)i , u

(2)i be the vertices of

ith copy of K2 attached with vi, for 1 ≤ i ≤ n. Define f : V (Pn ◦ K2) → {1, 2, 3, · · · , 4n} as

follows:

f(vi) = 4i− 2, for 1 ≤ i ≤ n,

f(u(1)i ) = 4i− 3, for 1 ≤ i ≤ n,

f(u(2)i ) = 4i, for 1 ≤ i ≤ n.


f∗(vivi+1) = 4i, for 1 ≤ i ≤ n− 1,

f∗(u(1)i u

(2)i ) = 4i− 2, for 1 ≤ i ≤ n,

f∗(u(1)i v1) = 4i− 3, for 1 ≤ i ≤ n,

f∗(u(2)i vi) = 4i− 1, for 1 ≤ i ≤ n.

Hence f is an F -centroidal mean labeling of Pn ◦K2 for n ≥ 1. Thus the graph Pn ◦K2 is an

F -centroidal mean graph, for n ≥ 1. 2AAAAAAs s AAAAAAs s AAAAAAs s s s AAAAAAs s2 6 10 14 18

1 4 5 8 9 12 13 1617 20

1 3 5 7 9 11 13 15 17 19

4 8 12 16 ss s s s6 10 14 182

Figure 6 An F -centroidal mean labeling of P5 ◦K2

Theorem 2.7 The graph P ba is an F -centroidal mean graph, for a ≥ 2 and b ≤ 3.

Proof Let yi, xij1, xij2, . . . xiji, yi+1 be the vertices of the path P ji , where 1 ≤ i ≤ a−1 and

1 ≤ j ≤ b. Let V (P ba) = {yi : 1 ≤ i ≤ a} ∪

a−1⋃i=1

b⋃j=1

{xijk : 1 ≤ k ≤ i} and E(P ba) =

a−1⋃i=1

{yixij1 :

1 ≤ i ≤ b} ∪a−1⋃i=1

b⋃j=1

{xijkxij(k+1) : 1 ≤ k ≤ i− 1} ∪a−1⋃i=1

{xijiyi+1 : 1 ≤ j ≤ b} be the vertex set

and edge set of the graph P ba .

Case 1. b = 2.

Define f : V (P 2a ) → {1, 2, 3, · · · , (a− 1)(a+ 2) + 1} as follows:

f(y1) = 1,

f(yi) = (i− 1)(i+ 2) + 1, for 2 ≤ i ≤ a,

f(x1j1) = j + 1, for 1 ≤ j ≤ 2 and for 2 ≤ i ≤ a− 1,


f(xijk) = (i− 1)(i+ 2) + 2k + j − 1, for 1 ≤ k ≤ i and 1 ≤ j ≤ 2.


f∗(y1x1j1) = j, for 1 ≤ j ≤ 2,

f∗(x1j1y2) = j + 2, for 1 ≤ j ≤ 2,

f∗(yixij1) = (i− 1)(i+ 2) + j, for 2 ≤ i ≤ a− 1 and 1 ≤ j ≤ 2,

f∗(xijkxij(k+1)) = (i− 1)(i+ 2) + j + 2k, for 2 ≤ i ≤ a− 1, 1 ≤ k ≤ i− 1 and 1 ≤ j ≤ 2,

f∗(xijiyi+1) = i(i+ 3) + j − 2, for 2 ≤ i ≤ a− 1 and 1 ≤ j ≤ 2.

Case 2. b = 3.

Define f : V (P 3a ) → {1, 2, 3, · · · , 3(a−1)(a+2)

2 + 1} as follows:

f(y1) = 1, f(y2) = 5, f(x111) = 2, f(x1j1) = 4j − 5, for 2 ≤ j ≤ 3,

f(yi) =3(i− 1)(i+ 2)

2+ 1, for 3 ≤ i ≤ a, f(x21k) =

4k + 5, k = 1,

4k + 4, k = 2,

f(x22k) =

7k + 6, k = 1,

7k − 4, k = 2,f(x23k) =

5k + 6, k = 1,

5k + 4, k = 2.

For 3 ≤ i ≤ a− 1,

f(xij1) =

3(i−1)(i+2)2 + j + 1, 1 ≤ j ≤ 2,

3(i−1)(i+2)2 + 2j, j = 3 and

f(xijk) =

3(i−1)(i+2)2 + 2j + 3k − 1, 1 ≤ j ≤ 2, 2 ≤ k ≤ i− 1,

and k is even,

3(i−1)(i+2)2 + 3k − 2, j = 3, 2 ≤ k ≤ i− 1

and k is even,

3(i−1)(i+2)2 + 2j + 3k − 3, 1 ≤ j ≤ 3, 2 ≤ k ≤ i− 1

and k is odd,3(i−1)(i+2)

2 + 3k − 1, j = 1, k = i and k is odd,3(i−1)(i+2)

2 + 3k + j − 1, j = 2, k = i and k is odd,3(i−1)(i+2)

2 + 3k + j, j = 3, k = i and k is odd,3(i−1)(i+2)

2 + 3k + j, 1 ≤ j ≤ 2, k = i,

and k is even,3(i−1)(i+2)

2 + 3k − 1, j = 3, k = i and k is even.



f∗(y1x1j1) =

j, 1 ≤ j ≤ 2,

5, j = 3,

f∗(yixij1) =

3(i−1)(i+2)2 + j, j = 1 and 2 ≤ i ≤ a− 1,

3(i−1)(i+2)2 + j + 1, j = 2 and i = 2,

3(i−1)(i+2)2 + j − 1, j = 3 and i = 2,

3(i−1)(i+2)2 + j, j = 2, 3 and 3 ≤ i ≤ a− 1,

f∗(x1j1y2) =

3, j = 1,

2j, 2 ≤ j ≤ 3,f∗(x2j2y3) =

14, j = 1,

13, j = 2,

15, j = 3,

f∗(x2j1x2j2) = j + 9 for 1 ≤ j ≤ 3 and 3 ≤ i ≤ a− 1,

f∗(xijkxij(k+1)) =

3(i−1)(i+2)2 + 3k + 2(j − 1) + 1, 1 ≤ k ≤ i− 1,

and 1 ≤ j ≤ 2,3(i−1)(i+2)

2 + 3k + 2, 1 ≤ k ≤ i− 1,

and j = 3,

and f∗(xijiyi+1) =

3i(i+3)2 + j − 3, 1 ≤ j ≤ 3 and i is odd,

3i(i+3)2 + j − 2, 1 ≤ j ≤ 2 and i is even,

3i(i+3)2 − 2, j = 3 and i is even.

Hence f is an F -centroidal mean labeling of P ba for a ≥ 2 and b ≤ 3. Thus the graph P b

a is an

F -centroidal mean graph, for a ≥ 2 and b ≤ 3. 21

2

3

8

11

12

14

16

17

15

13

1 3

2 4

13 15

17

12

14 16

18

5

7

6

11

195

79

68

10

s s s s s sss9

s s s sssss

s ss1

2

7

3

9 12

11 14

13 10

17 22 23

21 19 27

18 24 259 11 13 17 21 24 262 4

1 3

5 6

710

14

8

12

15

1619 22

18

20 23

27

25

s s ss ss s 28s5 16

s ss ss s ss s ssFigure 7 An F -centroidal mean labeling of P 2

4 and P 34


Theorem 2.8 The middle graph M(Pn) of a path Pn is an F -centroidal mean graph.

Proof Let V (Pn) = {v1, v2, v3, · · · , vn} and E(Pn) = {ei = vivi+1 : 1 ≤ i ≤ n− 1} be the

vertex set and edge set of the path Pn. Then,

V (M(Pn)) = {v1, v2, v3, . . . , vn, e1, e2, e3, · · · , en−1},E(M(Pn)) = {viei, eivi+1 : 1 ≤ i ≤ n− 1} ∪ {eiei+1 : 1 ≤ i ≤ n− 2}.

Define f : V (M(Pn)) → {1, 2, 3, · · · , 3n− 3} as follows:

f(vi) =

1, for i = 1,

3i− 3, for 2 ≤ i ≤ n,

f(ei) = 3i− 1, for 1 ≤ i ≤ n− 1.


f∗(viei) = 3i− 2, for 1 ≤ i ≤ n− 1,

f∗(eivi+1) = 3i− 1, for 1 ≤ i ≤ n− 1,

f∗(eiei+1) = 3i, for 1 ≤ i ≤ n− 2.

Hence f is an F -centroidal mean labeling of the graph M(Pn). Thus the graph M(Pn) is an

F -centroidal mean graph. 26

uuuu uu1 3 9 12

2 5 8 11

12 4 5

78

10 11u63 9

u uFigure 8 An F -centroidal mean labeling of M(P5)

Theorem 2.9 The total graph T (Pn) of a path Pn is an F -centroidal mean graph for n ≥ 1.

Proof Let V (Pn) = {v1, v2, v3, · · · , vn} and E(Pn) = {ei = vivi+1 : 1 ≤ i ≤ n− 1} be the

vertex set and edge set of the path Pn. Then V (T (Pn)) = {v1, v2, v3, · · · , vn, e1, e2, e3, · · · , en−1}and E(T (Pn)) = {vivi+1, eivi, eivi+1 : 1 ≤ i ≤ n− 1} ∪ {eiei+1 : 1 ≤ i ≤ n− 2}.

Define f : V (T (Pn)) → {1, 2, 3, . . . , 4(n− 1)} as follows:

f(v1) = 1,

f(vi) = 4i− 4, for 2 ≤ i ≤ n,

f(ei) = 4i− 2, for 1 ≤ i ≤ n− 1.



f∗(vivi+1) = 4i− 2, for 1 ≤ i ≤ n− 1,

f∗(eiei+1) = 4i, for 1 ≤ i ≤ n− 2,

f∗(viei) = 4i− 3, for 1 ≤ i ≤ n− 1,

f∗(eivi+1) = 4i− 1, for 1 ≤ i ≤ n− 1.

Hence f is an F -centroidal mean labeling of the graph T (Pn). Thus the graph T (Pn) is an

F -centroidal mean graph. 21 4 12 16 208

2 6 10 14 18

2 6 10 14 18

84 12 16

1 5 9 13 17

3 7 11 15 19

t t ttt t t ttt tFigure 9 An F -centroidal mean labeling of T (P6)

Theorem 2.10 The square graph P 2n of the path Pn is an F -centroidal mean graph for n ≥ 1.

Proof Let v1, v2, v3, · · · , vn be the vertices of the path Pn.Define f : V(P 2

n

)→ {1, 2, 3, · · · , 2(n−

1)} as follows:

f(v1) = 1,

f(vi) = 2i− 2, for 2 ≤ i ≤ n.


f∗(vivi+1) = 2i− 1, for 1 ≤ i ≤ n− 1,

f∗(vivi+2) = 2i, for 1 ≤ i ≤ n− 2.

Hence f is an F -centroidal mean labeling of the graph P 2n . Thus the graph P 2

n is an F -centroidal

mean graph. 21

2

4

6

8

1012

1 3 5 7 9 11

2 106

4 8t t t t tttFigure 10 An F -centroidal mean labeling of P 2

7


Theorem 2.11 The splitting graph S′(Pn) is an F -centroidal mean graph for n ≥ 2.

Proof Let v1, v2, · · · , vn be the vertices of the path Pn. Let v1, v2, · · · , vn, v′1, v

′2, · · · , v′n

be the vertices of the graph S′(Pn). Let V (S′(Pn)) = {vi, v′i : 1 ≤ i ≤ n} and E(S′(Pn)) =

{vivi+1, viv′i+1, v

′ivi+1 : 1 ≤ i ≤ n − 1} be the vertex set and edge set of the splitting graph

S′(Pn).

Case 1. n is odd.

Define f : V (S′(Pn)) → {1, 2, 3, · · · , 3n− 2} as follows:

f(vi) =

4i− 3, 1 ≤ i ≤ 2,

3, i = 3,

3i− 4, 4 ≤ i ≤ n and i is odd,

3i, 4 ≤ i ≤ n and i is even,

f(v′i) =

6, i = 1,

2, i = 2,

3i− 2, 3 ≤ i ≤ n.


f∗(vivi+1) =

i+ 2, 1 ≤ i ≤ 2,

3i− 1, 3 ≤ i ≤ n− 1,

f∗(viv′i+1) =

5i− 4, 1 ≤ i ≤ 2,

3i− 2, 3 ≤ i ≤ n− 1 and i is odd,

3i, 3 ≤ i ≤ n− 1 and i is even,

f∗(v′ivi+1) =

5, i = 1,

2, i = 2,

3i, 3 ≤ i ≤ n− 1 and i is odd,

3i− 2, 3 ≤ i ≤ n− 1 and i is even.

Case 2. n is even.

Define f : V (S′(Pn)) → {1, 2, 3, · · · , 3n− 2} as follows:

f(vi) =

4 − i, 1 ≤ i ≤ 2,


3i− 3, 3 ≤ i ≤ n and i is even,


f(v′i) =

1, i = 1,


3i− 2, 2 ≤ i ≤ n and i is even.


f∗(vivi+1) = 3i− 1, for 1 ≤ i ≤ n− 1,

f∗(viv′i+1) =

3i, 3 ≤ i ≤ n− 1 and i is odd,

3i− 2, 3 ≤ i ≤ n− 1 and i is even,

f(v′ivi+1) =

3i− 2, 1 ≤ i ≤ n− 1 and i is odd,

3i, 1 ≤ i ≤ n− 1 and i is even.

Hence f is an F -centroidal mean labeling of S′(Pn). Thus the splitting graph S′(Pn) is an

F -centroidal mean graph for n ≥ 2. 2

�� 1 4 6 10 12 16 18

22

3 2 8 9 14 15 20 212 5 8 11 14 17 20

3 4 9 10 15 16 21

1 6 7 12 13 18 19

��1 5 3 12 11 18

6 2 7 1013

16 19

3 4 8 14

1

5 2 9 10 15 16

6 712 13 18

171711

s s sss ss s s s s ss s

s s s ss s s s s ssss s s s

Figure 11 An F -centroidal mean labeling of S′(P7) and S′(P8)

Theorem 2.12 The graph P (1, 2, · · · , n− 1) is an F -centroidal mean graph for n ≥ 2.

Proof Let v1, v2, · · · , vn be the vertices of the path Pn and let uij be the vertices of

the partition of K2,miwith cardinality mi, 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ mi. Define f :

V (P (1, 2, · · · , n− 1)) → {1, 2, 3, · · · , n(n− 1) + 1} as follows:

f(vi) = i(i− 1) + 1, for 1 ≤ i ≤ n,

f(uij) = i(i− 1) + 2j, for 1 ≤ j ≤ i, and 1 ≤ j ≤ n− 1.



f∗(viuij) = i(i− 1) + j, for 1 ≤ j ≤ i and 1 ≤ i ≤ n− 1,

f(uijvi+1) = i2 + j, for 1 ≤ j ≤ i and 1 ≤ i ≤ n− 1.

Hence f is an F -centroidal mean labeling of P (1, 2, . . . , n−1). Thus the graph P (1, 2, . . . , n−1)

is an F -centroidal mean graph for n ≥ 2. 2AAAA SSSSq q q qq1

2

3

4 8

14

20

18

7 13 21

1 2 3 75 10 13

17

6 12

96 12 15 19

20

16

14188 11

q qqq22

31

30

28

26

2421

22

26

27

3025

24 2916

23 28

4

q qq q qqqqq

BBBBBBqq10q

Figure 12 An F -centroidal mean labeling of P (1, 2, 3, 4, 5)

References

[1] S.Arockiaraj, A.Durai Baskar and A.Rajesh Kannan, F -root square mean labeling of

graphs obtained from paths, International Journal of Mathematical Combinatorics, Vol.2

(2017), 92–104.

[2] S.Arockiaraj, A.Durai Baskar and A.Rajesh Kannan, F -root square mean labeling of some

cycle related graphs, Journal of Engineering Technology, 6 (2017), 440–448.

[3] A.Durai Baskar and P.Manivannan, F -heronian mean labeling of graphs obtained from

paths, International Journal of Pure and Applied Mathematics, 117(5) (2017), 55–62.

[4] A.Durai Baskar, S.Arockiaraj and B.Rajendran, Geometric meanness of graphs obtained

from paths, Util. Math., 101(2016), 45–68.

[5] A.Durai Baskar and S.Arockiaraj, F -geometric mean graphs, International Journal of Ap-

plications and Applied Mathematics, 10(2) (2015), 937–952.

[6] A.Durai Baskar and S.Arockiaraj, F -harmonic mean graphs, International Journal of

Mathematical Archieve, 5(8) (2014), 92–108.

[7] J.A.Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combina-

torics, 17(2017), #DS6.

[8] F.Harary, Graph Theory, Narosa Publishing House Reading, New Delhi, 1988.


Further Results on Analytic Odd Mean Labeling of Graphs

P.Jeyanthi1,R.Gomathi2 and Gee-Choon Lau3

1. Research Centre, Department of Mathematics

Govindammal Aditanar College for Women, Tiruchendur 628215, Tamil Nadu, India

2. Research Scholar, Reg.No.11825

Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627 012, Tamil Nadu, India

3. Faculty of Computer and Mathematical SciencesUniversiti Tek nologi MARA (Segamat Campus), 85009, Johor, Malaysia


Abstract: Let G = (V, E) be a graph with p vertices and q edges. A graph G is analytic

odd mean if there exist an injective function f : V → {0, 1, 3, 5 · · · , 2q − 1} with an induce

edge labeling f∗ : E → Z such that for each edge uv with f(u) < f(v),

f∗(uv) =

⌈f(v)2−(f(u)+1)2

2

⌉, if f(u) 6= 0

⌈f(v)2

2

⌉, if f(u) = 0

is injective. We say that f is an analytic odd mean labeling of G. In this paper we prove

that sun graph Sn, prism Dn, helm graph Hn, the graph Cn ◦P2, banana tree, bamboo tree,

perfect binary tree, the graph PCn, unicyclic graph, the caterpillar Pk(n0, n1, · · · , nk−1) and

spider graph are analytic odd mean graph.

Key Words: Mean labeling, analytic mean labeling, analytic odd mean labeling, Smaran-

dachely analytic odd mean labeling, analytic odd mean graph.

AMS(2010): 05C78.

§1. Introduction

Throughout this paper we consider only finite, simple and undirected graph G = (V,E) with

p vertices and q edges and notations not defined here are used in the sense of Harary [2].

A graph labeling is an assignment of integers to the vertices or edges or both, subject to

certain conditions. There are several types of labeling. An excellent survey of graph labeling

is available in [1]. The concept of analytic mean labeling was introduced in [7]. A graph

G is analytic mean graph if it admits a bijection f : V → {0, 1, 2, · · · , p− 1} such that the

induced edge labeling f∗ : E → Z given by f∗(uv) =⌈

f(u)2−f(v)2

2

⌉with f(u) > f(v) is

injective. Motivated by the results in [7], we introduced a new mean labeling called analytic

odd mean labeling in [3]. A graph G is an analytic odd mean if there exist an injective function

f : V → {0, 1, 3, 5 · · · , 2q − 1} with an induce edge labeling f∗ : E → Z such that for each edge


Further Results on Analytic Odd Mean Labeling of Graphs 137

uv with f(u) < f(v),

f∗(uv) =

⌈f(v)2−(f(u)+1)2

2

⌉, if f(u) 6= 0

⌈f(v)2

2

⌉, if f(u) = 0

is injective. We say such an f is an analytic odd mean labeling of G. Otherwise, a Smaran-

dachely analytic odd mean labeling of G if there exists an integer 0 ≤ k ≤ q holding with

|f−∗(k)| ≥ 2, where f−∗ is the inverse of f∗. We proved that cycle Cn, path Pn, n-bistar, comb

Pn ⊙K1, graph Ln ⊙K1, wheel graph Wn, flower graph Fln, some splitting graphs, multiple of

graphs, quadrilateral snake Q(n), double quadrilateral snake DQ(n), coconut tree, fire cracker

and some star graphs, splitting graph spl(G), Pn(1, 2, 3, · · ·, n), the complete bipartite graph

Km,n, the graph Ck⊙Kn, the square graph of Pn, Cn, Bn,n, H-graph and H⊙mK1 are analytic

odd mean graphs in [4], [5] and [6].

We use the following definitions in the subsequent section to prove the results.

Definition 1.1 A sun graph Sn is a cycle Cn with a pendent edge attached to each vertex of a

cycle Cn.

Definition 1.2 The prism Dn, n ≥ 3 is a cubic graph which can be represented as a Cartesian

product P2 × Cn of a path on two vertices with a cycle on n vertices.

Definition 1.3 A banana tree is a tree obtained from a family of stars by joining one end

vertex of each star to a new vertex.

Definition 1.4 A tree is called a spider if it has a center vertex c of degree R > 1 and all the

other vertex is either a leaf or with degree 2. Thus a spider is an amalgamation of k paths with

various lengths. If it has x1’s path of length a1, x2’s path of length a2,· · · . We denote the spider

graph by SP (a1x1 , a2

x2 , · · · , amxm) where a1 < a2 < · · · < am and x1 + x2 + · · · + xm = R.

Definition 1.5 A helm Hn, n ≥ 3 is obtained from the wheel graph Wn by adding a pendent

edge at each vertex on the wheel’s rim.

Definition 1.6 A bamboo tree (Pn ⊗ K1,ni)ki=1 is an one point union of Pn ⊗ K1,ni

where

1 ≤ i ≤ k.

Definition 1.7 A caterpillar Pk(n1, n2, · · · , nk) is a tree in which all the vertices are within

distance 1 of a central path Pk for k ≥ 1. When k ≥ 2, a caterpillar is obtained from a path

Pk = u1u2u3 · · ·uk attaching ni ≥ 0 pendent vertices vji (1 ≤ j ≤ ni) to each ui.

Definition 1.8 A perfect binary tree is a full binary tree in which all the leaves are at the same

level and in which every parent has two children.

§2. Main Results

In this section we prove that sun graph Sn, prism Dn, helm graph Hn, path union of n − 3

138 P.Jeyanthi, R.Gomathi and Gee-Choon Lau

copies of Cn,the graph Cn ◦ P2, banaba tree, bamboo tree, the graph PCn, unicyclic graph,

perfect binary tree, the caterpillar graph Pk(n0, n1, · · · , nk−1) and spider graph are analytic

odd mean graphs.

Theorem 2.1 For every positive integer n ≥ 3, the sun graph Sn is an analytic odd mean

graph.

Proof Let the vertex set and edge set of the sun graph be V (Sn) = {ui, vi : 1 ≤ i ≤ n}and E(Sn) = {uiui+1 : 1 ≤ i ≤ n− 1}⋃ {uivi : 1 ≤ i ≤ n}⋃ {unu1}Now |V (G)| = 2n = |E(G)|. We define an injective map f : V (Sn) → {0, 1, 3, 5, · · · , 4n− 1} by

f(ui) = 2i− 1 for 1 ≤ i ≤ n and f(vi) = 2n+ 2i− 1 for 1 ≤ i ≤ n.

The induced edge labeling f∗ is defined as follows:

f∗(uiui+1) = 2i+ 1 for 1 ≤ i ≤ n− 1,

f∗(u1un) = 2n2 − 2n− 1,

f∗(uivi) = 2n(n− 1) + 2i(2n− 1) + 1 for 1 ≤ i ≤ n.

We observe that the edge labels of uiui+1 are 3, 5, · · · 2n − 1 as i increases and the edge

labels of uivi are increased by 4n− 2 as i increases from 1 to n. Hence all the edge labels are

distinct and odd . Hence Sn admits an analytic odd mean labeling. 2An analytic odd mean labeling of S8 is shown in Figure 1.

b b

b

b

b

b

b b

b

b

b

b

b

b

b

b1

3

5

79

11

13

15

17

3

21

2325

27

29

31

5

7

9

11

13

15

111

143

173

203

233263

293

323

353

19

Figure 1

Theorem 2.2 For every positive n ≥ 3, the prism Dn is an analytic odd mean graph.

Proof Let the vertex set and edge set of the prism be V (Dn) = {ui, vi : 1 ≤ i ≤ n} and

E(Dn) = {uiui+1, vivi+1 : 1 ≤ i ≤ n−1}⋃{uivi : 1 ≤ i ≤ n}⋃ {unv1, vnv1}. Now |V (Dn)| =

2n and |E(Dn)| = 3n. We define an injective map f : V (Dn) → {0, 1, 3, 5, · · · , 6n − 1} by

f(ui) = 2i− 1 for 1 ≤ i ≤ n and f(vi) = 2n+ 2i− 1 for 1 ≤ i ≤ n.



f∗(uiui+1) = 2i+ 1 for 1 ≤ i ≤ n− 1,

f∗(u1un) = 2n2 − 2n− 1,

f∗(uivi) = 2n(n− 1) + 2i(2n− 1) + 1 for 1 ≤ i ≤ n,

f∗(vivi+1) = 2n+ 2i+ 1 for 1 ≤ i ≤ n− 1,

f∗(v1vn) = 6n2 − 8n− 1.

We observe that the edge labels of uiui+1 and vivi+1 are 3, 5, . . . , 2n− 1 and 2n+ 3, 2n+

5, · · · , 2n+ 2n− 1 = 4n− 1 respectively as i increases and the edge labels of uivi are increased

by 4n − 2 as i increases from 1 to n. So all the edge labels are distinct and odd. Hence Dn

admits an analytic odd mean labeling. 2An analytic odd mean labeling of D8 is shown in Figure 2.

b b

b

b

b

b

b b

b

b

b

b

b

b

b

b1

3

5

79

11

13

15

17

3

21

2325

27

29

31

5

7

9

11

13

15

111

143

173

203

233263

293

323

353

19

19

21

23

25

27

29

31

319

Figure 2

Theorem 2.3 The helm graph Hn, n ≥ 3 is an analytic odd mean graph.

Proof Let V (Hn) = {v, vi, ui : 1 ≤ i ≤ n} and E(Hn) = {vvi, viui : 1 ≤ i ≤ n} ∪{vivi+1 : 1 ≤ i ≤ n − 1} ∪ {vnv1}. Now |V (G)| = 2n + 1 and |E(G)| = 3n. We define an

injective map f : V (G) → {0, 1, 3, 5, · · · , 6n− 1} by f(v) = 0, f(vi) = 4i− 3 for 1 ≤ i ≤ n and

f(ui) = 4n+ 2i− 1 for 1 ≤ i ≤ n.


f∗(vvi) = 8i2 − 12i+ 5 for 1 ≤ i ≤ n,

f∗(vivi+1) = 12i− 1 for 1 ≤ i ≤ n− 1,

f∗(vnv1) = 8n2 − 12n+ 3,

f∗(viui) = 4n(2n− 1) − 6i(i− 1) + 8ni− 1 for 1 ≤ i ≤ n.


We observe that when i increases, the difference of edges viui are decreased by 12. Clearly

all the edge labels are odd and distinct. Hence Hn admits an analytic odd mean labeling. 2An analytic odd mean labeling of H10 is shown in Figure 3.

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b b

b

b

b

v

v9

v8

v7

v6

v5

v4

v3

v2

v1

u10

u9

u8

u7

u6

u5

u4

u3

u2

u1

v10

0

57

53

51

49

47

45

43

41

37

33

29

2521

17

13

9

5

1

55

59

Figure 3

Theorem 2.4 The graph PCn(n ≥ 4 and n is even) is obtained from Cn = v1v2 · · · vnv1 by

adding the chords vi and vn−i+2 for 2 ≤ i ≤ l where l = n/2. Then the graph PCn is an

analytic odd mean graph.

Proof Let G = PCn. Let the vertex set and edge set of G be V (G) = {vi, : 1 ≤ i ≤ n}and E(G) = {vivi+1 : 1 ≤ i ≤ n − 1} ∪ {vivn−i+2 : 2 ≤ i ≤ l} ∪ {vnv1}. Then there are

n vertices and 3n/2 − 1 edges. We define an injective map f : V (G) = {0, 1, 3, · · ·3n− 3} by

f(vi) = 2i− 1 for 1 ≤ i ≤ n.


f∗(vivi+1) = 2i+ 1 for 1 ≤ i ≤ n,

f∗(vnv1) = 2n2 − 2n− 1,

f∗(vivn−i+2) = 2n(n+ 3) − 2i(2n+ 3) + 5 for 2 ≤ i ≤ l.

It can be easily verified that f is an analytic odd mean labeling and hence PCn is an

analytic odd mean graph. 2An analytic odd mean labeling of PC10 is shown in Figure 4.


b

b

b

b

b

b

b

b

b

b

1

3

5

7

9

11

13

15

17

19 173

127

81

35

179 3

5

7

9

1113

15

17

19

Figure 4

Theorem 2.5 Let G be a unicycle graph with a cycle Ck = a1a2 · · · aka1 such that the vertex

ai is attached to ni pendent vertices. Then the unicycle graph admits an analytic odd mean

labeling.

Proof Let V (G) = {ai, aji : 1 ≤ i ≤ k and 1 ≤ j ≤ ni} and E(G) = {aiai+1 : 1 ≤

i ≤ k − 1} ∪ {aiaji : 1 ≤ i ≤ k and 1 ≤ j ≤ ni} ∪ {aka1}. We assume n0 = 0. Hence

|V | = n1 + n2 + · · · + nk + k = |E| .We define an injective map on the vertex set by f(ai) = 2i − 1 for 1 ≤ i ≤ k and

f(aji ) = 2k − 1 + 2

∑i−1r=0 nr + 2j for 1 ≤ i ≤ k and 1 ≤ j ≤ ni.


f∗(aiai+1) = 2i+ 1 for 1 ≤ i ≤ k − 1,

f∗(aka1) = 2k2 − 2k − 1,

f∗(aiaji ) = 2(k2 + j2 − i2) + 2(

∑i−1r=0 nr)

2 + 2(2k+ 2j − 1)∑j−1

i=1 ni + 4kj − 2(k+ j) + 1 for

1 ≤ i ≤ k and 1 ≤ j ≤ ni.

Clearly the edge labels are odd and distinct. Hence the unicycle graph admits an analytic

odd mean labeling. 2An analytic odd mean labeling of the unicycle graph with k = 6 is shown in Figure 5.

b

b b

b b

b

b b

b

bb

b

b

b

b

b

b

b

b

b

b

b

b

1

43

41

39

35

37

33

11

9

7

5

31 29 27 25

21

23

19

17

13 15

45

3

3

5

79

11

83 111

137

163

203

333281

247563

495

449389

853

769689

613

941

Figure 5

59


Theorem 2.6 The caterpillar Pk(n0, n1, · · · , nk−1) is an analytic odd mean graph for k ≥2, ni ≥ 0.

Proof Let V (G) = {vi, vji : 0 ≤ i ≤ k − 1 and 1 ≤ j ≤ ni} and E(G) = {vi−1vi : 1 ≤

i ≤ k − 1} ∪ {vivji : 0 ≤ i ≤ k − 1 and 1 ≤ j ≤ ni}. Hence |V | = n0 + n1 + . . . + nk + k and

|E| = n0 + n1 + · · · + nk + k − 1. We define an injective map on the vertex set by

f(v0) = 0, f(vi) = 2i− 1 for 1 ≤ i ≤ k − 1,

f(vj0) = 2k + 2j − 3 for 1 ≤ j ≤ ni,

f(vji ) = 2k − 3 + 2

∑i−1r=0 nr + 2j for 1 ≤ i ≤ k − 1 and 1 ≤ j ≤ ni.


f∗(vi−1vi) = 2i− 1 for 1 ≤ i ≤ k − 1,

f∗(v0vj0) = 2k(k − 3) + 2j(j − 3) + 4kj + 5 for 1 ≤ j ≤ ni,

f∗(vivji ) =

[(2k − 3 + 2

∑i−1r=0 nr + 2j − 2i

)(2k − 3 + 2

∑i−1r=0 nr + 2j + 2i

)+ 1]÷ 2 for

1 ≤ i ≤ k − 1 and 1 ≤ j ≤ ni.

Clearly the edge labels are odd and distinct. Hence Pk(n0, n1, ..., nk−1) admits an analytic

odd mean labeling. 2An analytic odd mean labeling of P5(3, 2, 1, 5, 4) is shown in Figure 6.

bb b b b

b

b

b b b b b

b bb

b b

b b

b

3119171513

119

75310

332927

2523

21 3537

Figure 6

Theorem 2.7 The graph Cn ◦ P2 is an analytic odd mean graph.

Proof Let G be the graph Cn ◦ P2. Let V (G) = {vi, vi,1, vi,2 : 1 ≤ i ≤ n} and E(G) =

{vivi+1 : 1 ≤ i ≤ n− 1} ∪ {vnv1} ∪ {vivi,1, vivi,2, vi,1vi,2 : 1 ≤ i ≤ n}. Now |V (G)| = 3n and

|E(G)| = 4n.

We define an injective map f : V (G) → {0, 1, 3, 5, · · · , 8n− 1} by

f(vi) = 2i− 1 for 1 ≤ i ≤ n,

f(vi,1) = 4i+ 2n− 3 for 1 ≤ i ≤ n,

f(vi,2) = 4i+ 2n− 1 for 1 ≤ i ≤ n.


f∗(vivi+1) = 2i+ 1 for 1 ≤ i ≤ n− 1,

f∗(vnv1) = 2n2 − 2n+ 1,

f∗(vivi,1) = 6i(i− 2) + 2n(n− 3) + 8ni+ 5 for 1 ≤ i ≤ n,

f∗(vivi,2) = 2i(3i− 2) + 2n(n− 1) + 8ni+ 1 for 1 ≤ i ≤ n,

f∗(vi,1vi,2) = 4i+ 2n− 1 for 1 ≤ i ≤ n.


Clearly the edge labels are odd and distinct. Hence Cn ◦ P2 admits an analytic odd mean

labeling. 2An analytic odd mean labeling of C5 ◦ P2 is shown in Figure 7.

b b

b b

b

b

b

b

b

bb

b

b b

b

21

17

13

29

27

25

23 21

19

17

15

1311

39

9

7

5

3

9

7 5

3

1

203

163

25

233

281

315

37129

59 83

105

137

Figure 7

Theorem 2.8 Let K1,n1 ,K1,n2 , · · · ,K1,nkbe a family of stars with vertex sets V (K1,nj

) ={aj , a

1j , a

2j , · · · , a

nj

j

}, and deg(aj) = nj , 1 ≤ j ≤ k. Let BT (n1, n2, · · · , nk) be a banana tree

obtained by adding a new vertex a and joning it to a11, a

12, a

13, · · · , a1

k. Then BT (n1, n2, · · · , nk)

admits an analytic mean labeling where nj is any positive integer.

Proof Let the vertex set and edge set be V ={a, aj, a

rj : 1 ≤ j ≤ k, 1 ≤ r ≤ nj

}and E ={

aa1j : 1 ≤ j ≤ k

}∪{aja

rj : 1 ≤ j ≤ k, 1 ≤ r ≤ nj

}respectively. We assume n0 = 0. Hence

|V | = n1 + n2 + · · · + nk + k + 1 and |E| = n1 + n2 + · · · + nk + k.

We define an injective function f on the vertex set of banana tree as follows :

f(a) = 0, f(aj) = 2j − 1 for 1 ≤ j ≤ k,

f(arj) = 2k − 1 + 2

∑j−1i=0 ni + 2r for 1 ≤ r ≤ nj , 1 ≤ j ≤ k.


f∗(ajarj) = 2(k2 + r2 − j2) + 2(

j−1∑

i=0

ni + j)(2k − 1 +

j−1∑

i=0

ni) + 2j

j−1∑

i=0

ni − 2k + 1

for 1 ≤ r ≤ nj , 1 ≤ j ≤ k and

f∗(aa1j) =

[(2k − 1 + 2

∑j−1i=0 ni + 2

)2

+ 1

]

2

= 2k(k + 1) + 2

j−1∑

i=0

ni(2k + 1) + 2(

j−1∑

i=0

ni)2 + 1


for 1 ≤ j ≤ k.

Clearly the edge labels are odd and distinct. Therefore f is an analytic odd mean labeling

and hence the banana tree is an analytic odd mean graph. 2An analytic odd mean labeling of BT (3, 5, 6, 4) is shown in Figure 8.

b

b b

b b

b

bb

b

b b

b

b

b

b

b

b

b

b

b

b

b

b

23

21

1917

13

9

11

7

53

1

0

2931

33

35

37

2515

105

43

41

39

27

257

213

173137

8359

685

313113

41

39

893

809

729653

595

527

463403

347

295

Figure 8

Theorem 2.9 Let K1,n1 ,K1,n2 , · · · ,K1,nk,K1,k be a family of stars with vertex sets V (K1,nj

) ={vr

j : 0 ≤ r ≤ nj

}and deg(v0

j ) = nj , for 1 ≤ j ≤ k, V (K1,k) = {v, v1, v2, · · · , vk}. The bam-

boo tree (P2 ⊗ K1,nj) for j = 1, 2, . . . , k is obtained by joining the vertex v1, v2, . . . , vk with

v11 , v

12 , · · · , v1

k respectively. Clearly the number of vertices and edges of the bamboo tree are

n1 + n2 + · · · + nk + k + 1 and n1 + n2 + . . . + nk + k respectively. Then the bamboo tree

(P2 ⊗K1,nj) for j = 1, 2, · · · , k is an analytic odd mean graph.

Proof Let the vertex set and edge set be V ={v, vj , v

rj : 1 ≤ j ≤ k, 0 ≤ r ≤ nj

}and

E = {vvj : 1 ≤ j ≤ k}∪{vjv

1j : 1 ≤ j ≤ k

}∪{v0

j vrj : 1 ≤ j ≤ k, 1 ≤ r ≤ nj

}respectively. We

assume n0 = 0. Hence |V | = n1 + n2 + . . .+ nk + 2k + 1 and |E| = n1 + n2 + · · · + nk + 2k.

We define an injective function f on the vertex set of bamboo tree as follows:

f(v) = 0, f(vj) = 2j − 1 for 1 ≤ j ≤ k,

f(vrj ) = 4k − 1 + 2

∑j−1i=0 ni + 2r for 1 ≤ r ≤ nj , 1 ≤ j ≤ k,

f(v0j ) = 2k + 2j − 1 for 1 ≤ j ≤ k.


f∗(vvj) = 2j2 − 2j + 1 for 1 ≤ j ≤ k,

f∗(vjv1j ) =

[(4k + 1 + 2

∑j−1i=0 ni

)2

+ 1

]÷ 2 − 2j2 for 1 ≤ j ≤ k,

f∗(v0j v

rj ) =

[(4k − 1 + 2

∑j−1i=0 ni + 2r

)2

+ 1

]÷ 2 − 2(k2 + j2) − 4kj for 1 ≤ r ≤ nj , 1 ≤

j ≤ k.

Clearly the edge labels are odd and distinct. Therefore f is an analytic odd mean labeling

and hence the bamboo tree is an analytic odd mean graph. 2


An analytic odd mean labeling of (P2 ⊗K1,nj) for j = 1, 2, 3, 4 with n1 = 5, n2 = 4, n3 =

3, n4 = 7 is shown in Figure 9.

bb

b

b

b

b

b b

b b b b

b

b

b

b

b

b

b

b

bb

b b

b

b

b

b

47

33

31

2925

23

21

19

9

41352717

7531

0

49

5153

151311

3937

43

45

v64v5

1

v41

v31

v21

v02v0

1

v14

v13v1

2v11

v4v3v2v1

v

v54

v74

v04

v33v2

3

v03

v32

v42

v22

v24

v34

v44

Figure 9

v

Theorem 2.10 The perfect binary tree T of order p is an analytic odd mean graph.

Proof Let V (T ) = {vi : 1 ≤ i ≤ p} and E(T ) = {viv2i, viv2i+1 : 1 ≤ i ≤ q/2}. Hence

|V | = p and |E| = p − 1. We define an injective map f : V (G) = {0, 1, 3, · · · , 2p− 3} by

f(v1) = 0 and f(vi) = 2i− 3 for 2 ≤ i ≤ p. The induced edge labeling f∗ is defined as follows:

f∗(v1v2) = 1,

f∗(v1v3) = 5,

f∗(viv2i) = 6i2 − 8i+ 3 for 2 ≤ i ≤ q/2,

f∗(viv2i+1) = 6i2 − 1 for 2 ≤ i ≤ q/2.

We observe that the difference of edge labels of viv2i and viv2i+1 is 8i − 4 as i increases

from 1 to q2 . Therefore the edge labels are odd and distinct. Hence the binary tree admits an

analytic odd mean labeling. 2An analytic odd mean labeling of T of order 15 is shown in Figure 10.

b

bb

b bb b

b b b b b b b b

0

19171513

11975

31

27252321

Figure 10

53

293241215171113 1499567

332311

1 5


Theorem 2.11 The spider graph SP (1n, km), n ≥ 1, and k,m ≥ 2 is an analytic odd mean

graph.

Proof Let V (SP (1n, km)) = {v} ∪ {wa, : 1 ≤ a ≤ n} ∪ {vji : 1 ≤ i ≤ k and 1 ≤ j ≤ m}

and E(SP (1n, km)) = {vvj1, v

ji v

ji+1 : 1 ≤ i ≤ k − 1 and 1 ≤ j ≤ m} ∪ {vwa : 1 ≤ a ≤ n}.

We define an injective map f : V (SP (1n, km)) → {0, 1, 3, 5, · · · , 2(km + n) − 1} by f(v) =

0, f(vji ) = (j − 1)2k+ 2i− 1 for 1 ≤ i ≤ k, 1 ≤ j ≤ m and f(wa) = 2mk+ 2a− 1 for 1 ≤ a ≤ n.


f∗(vvj1) = 2k(j − 1)(k(j − 1) + 1) + 1 for 1 ≤ j ≤ m,

f∗(vji v

ji+1) = 2k(j − 1) + 2i+ 1 for 1 ≤ i ≤ k − 1 and 1 ≤ j ≤ m,

f∗(vwa) = 2mk(mk − 1) + 2a(a− 1) + 4mak + 1 for 1 ≤ a ≤ n.

It can be verified that the edge labels are odd and distinct. Hence SP (1n, km) is an analytic

odd mean graph. 2An analytic odd mean labeling of SP (1n, 65) is shown in Figure 11.

b

b

b

b

b b b b b b

b b bb b

b b b bb

b b b b b

b bb b b

b

b

b

b

b

b

b

51

29 31 33 35

232119171513

1197531

0

49

474543413937

25 27

63

61

57 595553

2mk + 2n− 1

Figure 11

Theorem 2.12 The spider graph SP (2n, km), n ≥ 1, k,m ≥ 2 is an analytic odd mean graph

if (a) k is even and m is any integer and (b) k is odd and m is even.

Proof Let V (SP (2n, km)) = {v} ∪ {wa,1, wa,2 : 1 ≤ a ≤ n} ∪ {vji : 1 ≤ i ≤ k and 1 ≤ j ≤

m} and E(SP (2n, km)) = {vvj1, v

ji v

ji+1 : 1 ≤ i ≤ k ans 1 ≤ j ≤ m} ∪ {vwa,1, wa,1wa,2 : 1 ≤

a ≤ n}. We define an injective map f : V (SP (1n, km)) → {0, 1, 3, 5, · · · , 2km + 4n − 1} by

f(v) = 0, f(vji ) = (j − 1)2k + 2i − 1 for 1 ≤ i ≤ k, 1 ≤ j ≤ m; f(wa,1) = 2mk + 4a − 3 for

1 ≤ a ≤ n and f(wa,2) = 2mk + 4a− 1 for 1 ≤ a ≤ n.


f∗(vvj1) = 2k(j − 1)(k(j − 1) + 1) + 1 for 1 ≤ j ≤ m,

f∗(vji v

ji+1) = 2k(j − 1) + 2i+ 1 for 1 ≤ i ≤ k − 1 and 1 ≤ j ≤ m,

f∗(vwa,1) = 2mk(mk − 3) + 4a(2a− 3) + 8mak + 5 for 1 ≤ a ≤ n,

f∗(wa,1wa,2) = 2mk + 4a− 1 for 1 ≤ a ≤ n.


It can be verified that the edge labels are odd and distinct. Hence SP (2n, km) is an analytic

odd mean graph. 2An analytic odd mean labeling of SP (2n, 54) is shown in Figure 12.

b b b b b

bb b b b

b b b b b

b b b bb

b

b b

b b

b b

b

b

b

0

1 3 5 7 9

11 13 15 17 19

21 23 25 27 29

31 33 35 37 39

41 43

45 47

2mk + 4n− 12mk + 4n− 3

Figure 12

Theorem 2.13 The spider graph SP (1s, 2n, km), n, s ≥ 1, k,m ≥ 2 is an analytic odd mean

graph if (a) k is even and m is any integer and (b) k is odd and m is even.

Proof Let V (SP (1s, 2n, km)) = {v} ∪ {wa,1, wa,2 : 1 ≤ a ≤ n} ∪ {vji : 1 ≤ i ≤ k

and 1 ≤ j ≤ m} ∪ {ur : 1 ≤ r ≤ s} and E(SP (1s, 2n, km)) = {vvj1, v

ji v

ji+1 : 1 ≤

i ≤ k and 1 ≤ j ≤ m} ∪ {vwa,1, wa,1wa,2 : 1 ≤ a ≤ n}. We define an injective map

f : V (SP (1n, km)) → {0, 1, 3, 5, · · · , 2km + 4n + 2s − 1} by f(ur) = 2mk + 4n + 2r − 1 for

1 ≤ r ≤ s and f(v), f(vji ), f(wa,1) and f(wa,2) are defined as in Theorem 2.12. Then the

induced edge labeling f∗(vvj1), f

∗(vji v

ji+1), f

∗(vwa,1) and f∗(wa,1wa,2) are as in Theorem 2.12

and

f∗(vur) =(2mk + 2r + 4n− 1)

2+ 1

2

for 1 ≤ r ≤ s.

It can be verified that the edge labels are odd and distinct. Hence SP (1s, 2n, km) is an

analytic odd mean graph. 2An analytic odd mean labeling of SP (1s, 2n, 34) is shown in Figure 13.

b b b

b b

b b

b

b

b

b

b

b

b

b

b

bb b

b b b

bb b

b0

1 3 5

7 9 11

13 15 17

19 21 23

25 27

2mk + 4n− 3 2mk + 4n− 1

2mk + 4n+ 2s− 1

2mk + 4n+ 3

2mk + 4n+ 1

Figure 13


References

[1] J.A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics,

(2018), # DS6.

[2] F.Harary, Graph Theory, Addison-Wesley ,Reading, Massachusetts, 1972.

[3] P.Jeyanthi, R.Gomathi and Gee-Choon Lau, Analytic odd mean labeling of square and

H-graphs, International J.Math. Combin., Special Issue 1 (2018), 61-67.

[4] P.Jeyanthi,R.Gomathi and Gee-Choon Lau, Analytic odd mean labeling of some standard

graphs, Palestine Journal of Mathematics, Vol.8(1)(2019), 444-450.

[5] P.Jeyanthi,R.Gomathi and Gee-Choon Lau, Some results on analytic odd mean labeling of

graph, Bulletin of The International Mathematical Virtual Institute, Vol. 9(2019), 487-500.

[6] P.Jeyanthi,R.Gomathi and Gee-Choon Lau, Analytic odd mean labeling of some graphs,

Palestine Journal of Mathematics, Vol.8(2)(2019), 392-399.

[7] T.Tharmaraj and P.B.Sarasija, Analytic mean labelled graphs, International Journal of

Mathematical Archive, 5(6)(2014), 136-146.

Papers Published in IJMC, 2019

Vol.1,2019

1. Harmonic Flows Dynamics on Animals in Microscopic Level with Balance Recovery

Linfan MAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01

2. Null Quaternionic Slant Helices in Minkowski Spaces

T.Kahraman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3. Unique Metro Domination Number of Circulant Graphs

B. Sooryanarayana and John Sherra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4. On Hemi-Slant Submanifold of Kenmotsu Manifold

Chhanda Patra, Barnali Laha and Arindam Bhattacharyya . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5. The Number of Rooted Nearly 2-Regular Loopless Planar Maps

Shude Long and Junliang Cai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6. A Note on Common Fixed Points for (ψ, α, β)-Weakly Contractive Mappings in Gener-

alized Metric Space

Krishnadhan Sarkar and Kalishankar Tiwary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7. Zk-Magic Labeling of Cycle of Graphs

P.Jeyanthi and K.Jeya Daisy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88

8. Topological Efficiency Index of Some Composite Graphs

K.Pattabiraman and T.Suganya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9. Total Domination Stable Graphs

Shyama M.P. and Anil Kumar V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

10. Centered Triangular Mean Graphs

P.Jeyanthi, R.Kalaiyarasi and D.Ramya. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126

11. New Families of Odd Mean Graphs

G.Pooranam, R.Vasuki and S.Suganthi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134

12. F -Root Square Mean Labeling of Some Graphs

R.Gopi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Vol.2,2019

1. Computing Zagreb Polynomials of Generalized xyz-Point-Line Transformation Graphs

T xyz(G) With z = −B. Basavanagoud and Anand P. Barangi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01

2. On (j,m) Symmetric Convex Harmonic Functions

Renuka Devi K, Hamid Shamsan and S. Latha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. On M-Projective Curvature Tensor of a (LCS)n-Manifold

Divyashree G. and Venkatesha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4. D-homothetic Deformations of Lorentzian Para-Sasakian Manifold

150 International Journal of Mathematical Combinatorics

Barnali Laha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5. Position Vectors of the Curves in Affine 3-Space According to Special Affine Frames

Yılmaz TUNCER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6. A Generalization of Some Integral Inequalities for Multiplicatively P-Functions

Huriye Kadakal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7. C-Geometric Mean Labeling of Some Cycle Related Graphs

A.Durai Baskar and S.Arockiaraj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8. Neighbourhood V4-Magic Labeling of Some Shadow Graphs

Vineesh K.P. and Anil Kumar V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

9. Open Packing Number of Triangular Snakes

S.K.Vaidya and A.D.Parmar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10. Second Status Connectivity Indices and its Coindices of Composite Graphs

K.Pattabiraman and A.Santhakumar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

11. A Note on Detour Radial Signed Graphs

K. V. Madhusudhan and S. Vijay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

12. A Note on 3-Remainder Cordial Labeling Graphs

R.Ponraj, K.Annathurai and R.Kala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Vol.3,2019

1. On the Involute D-scroll in Euclidean 3-Space

Seyda Kılıcoglu and Suleyman SENYURT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01

2. On Prime Graph PG2(R) of a Ring

Sandeep S.Joshi and Kishor F.Pawar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3. Non-Holonomic Frame for a Finsler Space with Some Special (α, β)-Metric

C.K.Yadav and M.K.Gupta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4. R1 Concepts in Fuzzy Bitopological Spaces with Quasi-Coincidence Sense

Ruhul Amin and Sahadat Hossain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5. Separations Axioms (T1) on Fuzzy Bitopological Spaces in Quasi-Coincidence Sense

Saikh Shahjahan Miah, M. R. Amin and Md.Fazlul Hoque . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6. On Radicals for Ternary Semirings

Swapnil P. Wani and Kishor F. Pawar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7. Computation of Misbalance Type Degree Indices of Certain Classes of Derived-Regular

Graph

V. Lokesha, K. Zeba Yasmeen, B. Chaluvaraju and T. Deepika . . . . . . . . . . . . . . . . . . . . . . . . 61

8. Reciprocal Transmission Hosoya Polynomial of Graphs

Harishchandra S. Ramane and Saroja Y. Talwar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

9. Generalized Common Neighbor Polynomial of Graphs

Shikhi M. and Anil Kumar V.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80

10. On Status Coindex Distance Sum and Status Connectivity Coindices of Graphs

Kishori P. Narayankar, Dickson Selvan and Afework T. Kahsay . . . . . . . . . . . . . . . . . . . . . . . 90

11. Some Results on 4-Total Prime Cordial Graphs

R.Ponraj, J.Maruthamani and R.Kala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Papers Published In IJMC, 2019 151

12. Super F -Centroidal Mean Graphs

S.Arockiaraj, A.Rajesh Kannan and A.Durai Baskar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Vol.4,2019

1. Graphs, Networks and Natural Reality – from Intuitive Abstracting to Theory

Linfan MAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01

2. Some Subclasses of Meromorphic with p-Valent q-Spirallike Functions

Read.S.A.Qahtan, Hamid Shamsan and S.Latha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. On the Complexity of Some Classes of Circulant Graphs and Chebyshev Polynomials

S. N. Daoud and Muhammad Kamran Siddiqui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4. Some Fixed Point Results for Multivalued Nonexpansive Mappings in Banach Spaces

G.S.Saluja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5. Algebraic Properties of the Path Complexes of Cycles

Seyed Mohammad Ajdani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6. Some Classes of Analytic Functions with q-Calculus

Hamid Shamsan, Fuad S. M. Alsarari and S. Latha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7. On Obic Algebras

Ilojide Emmanuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8. Prime BI-Idels of Po-Γ-Groupoids

J.Catherine Grace John and B. Elavarasan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

9. 4-Total Prime Cordiality of Certain Subdivided Graphs

R. Ponraj, J. Maruthamani and R. Kala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

10. The Signed Product Cordial for Corona of Paths and Fourth Power of Paths

S. Nada, A. Elrokh, A. Elrayes and A. Rabie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102

11. On the Eccentric Sequence of Composite Graphs

Yaser Alizadeh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

12. F -Centroidal Mean Labeling of Graphs Obtained From Paths

S. Arockiaraj, A. Rajesh Kannan and A. Durai Baskar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

13. Further Results on Analytic Odd Mean Labeling of Graphs

P.Jeyanthi, R.Gomathi and Gee-Choon Lau. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Famous Words

The reality of a thing T is the behavior with motivation of an abstracted complex network

in the microcosmic level. Certainly, there are more microcosmic observing datum on the units,

cells or microcosmic particles of matters by scientific instruments. A microcosmic science is such

a science established on the microcosmic datum of matters, including theory and experimental

subjects, which must be established over 1-dimensional skeleton or in other words, topological

graphs. Could we establish such a mathematics over topological graphs for microcosmic science?

The answer is positive inspired by the traditional Chinese medicine, i.e., 12 meridians theory.

(Extracted from the paper: Science’s Dilemma - a Review on Science with Applications,

Progress in Physics, Vol.15, 2(2019), 78-85.)

By Dr.Linfan MAO, a Chinese mathematician, philosophical critic.

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following style:

Books

[4]Linfan Mao, Combinatorial Geometry with Applications to Field Theory, InfoQuest Press,

2009.

[12]W.S.Massey, Algebraic topology: an introduction, Springer-Verlag, New York 1977.

Research papers

[6]Linfan Mao, Mathematics on non-mathematics - A combinatorial contribution, International

J.Math. Combin., Vol.3(2014), 1-34.

[9]Kavita Srivastava, On singular H-closed extensions, Proc. Amer. Math. Soc. (to appear).

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accepted for publication on the basis of the recommendation of referees. Corrections should be

restricted to typesetting errors. Authors are advised to check their proofs very carefully before

return.

December 2019

Contents

Graphs, Networks and Natural Reality –from Intuitive Abstracting to Theory

By Linfan MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01

Some Subclasses of Meromorphic with p-Valent q-Spirallike Functions

By Read.S.A.Qahtan, Hamid Shamsan and S.Latha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

On the Complexity of Some Classes of Circulant Graphs and Chebyshev

Polynomials By S. N. Daoud and Muhammad Kamran Siddiqui. . . . . . . . . . . . . . . . . . . . . . .29

Some Fixed Point Results for Multivalued Nonexpansive Mappings in Banach

Spaces By G.S.Saluja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Algebraic Properties of the Path Complexes of Cycles

By Seyed Mohammad Ajdani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Some Classes of Analytic Functions with q-Calculus

By Hamid Shamsan, Fuad S. M. Alsarari and S. Latha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

On Obic Algebras By Ilojide Emmanuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Prime BI-Ideals of Po-Γ-Groupoids

By J.Catherine Grace John and B. Elavarasan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4-Total Prime Cordiality of Certain Subdivided Graphs

By R. Ponraj, J. Maruthamani and R. Kala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

The Signed Product Cordial for Corona of Paths and Fourth Power of Paths

By S. Nada, A. Elrokh, A. Elrayes and A. Rabie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

On the Eccentric Sequence of Composite Graphs By Yaser Alizadeh . . . . . . . . . . . 112

F -Centroidal Mean Labeling of Graphs Obtained From Paths

By S. Arockiaraj, A. Rajesh Kannan and A. Durai Baskar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Further Results on Analytic Odd Mean Labeling of Graphs

By P.Jeyanthi, R.Gomathi and Gee-Choon Lau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Papers Published in IJMC, 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

An International Journal on Mathematical Combinatorics

Documents

fs.unm.edufs.unm.edu/IJMC/IJMC-4-2019.pdf · Aims and Scope: The mathematical combinatorics is a subject that applying combinatorial notion to all mathematics and all sciences for